On degree-based operators and topological descriptors of molecular graphs and their applications to QSPR analysis of carbon derivatives

Abstract

This work initiates a concept of reduced reverse degree based $\mathcal {RR_D\,M}$-Polynomial for a graph, and differential and integral operators by using this $\mathcal {RR_D\,M}$-Polynomial. In this study twelve reduced reverse degree-based topological descriptors are formulated using the $\mathcal {RR_D\,M}$-Polynomial. The topological descriptors, denoted as $\mathbb{T}\mathbb{D}$’s, are numerical invariants that offer significant insights into the molecular topology of a molecular graph. These descriptors are essential for conducting QSPR investigations and accurately estimating physicochemical attributes. The structural and algebraic characteristics of the graphene and graphdiyne are studied to apply this methodology. The study involves the analysis and estimation of Reduced reverse degree-based topological descriptors and physicochemical features of graphene derivatives using best-fit quadratic regression models. This work opens up new directions for scientists and researchers to pursue, taking them into new fields of study.

Introduction

The discovery of graphene has revolutionized materials science, igniting curiosity in additional 2D carbon allotropes like graphynes and grapevines. In 1987, these materials were initially predicted¹. The significance of two-dimensional graphene, graphyne, and graphdiyne derivatives of graphite structures is growing due to its encouraging characteristics, which include energy level alignment, charge carrier mobilities, and tunable band gaps^2,3. Two-dimensional carbon allotrope families with acetylenic groups joining benzenoid-like hexagonal rings are referred to as graphynes or graphdiynes⁴. The useful approach for forecasting drug-drug interactions based on knowledge graph neural networks and molecular substructures was presented by Chen et al.⁵.

Graphene is a two-dimensional in-nature sheet composed of hexagons and $sp^{2}$ hybridized carbon atoms⁶. Because carbon atoms are so adaptable, it is theoretically possible to create carbon allotropes by modifying the periodic patterns within networks of $sp^{3}$-, $sp^{2}$-, and sp-hybridized carbon atoms^7,8. The main distinction between graphynes and grapevines is the presence of one or two acetylenic groups. Increasing the amount of acetylenic groups can theoretically lead to an almost endless number of related structures. Graphene has attracted a lot of attention because of its remarkable mechanical, chemical, thermal, electrical, and physical properties^9,10.

Recently, graphene-related research has received a lot of attention because it was honoured with the 2010 Nobel Prize in Physics for its “groundbreaking relating to the two-dimensional (2D) substance”¹¹. When the bonds between the three coordinated atoms in a graphene layer are swapped with carbyne chains, the states of the $sp^{2}$ atoms stay equivalent and graphene layers are formed¹².

Graphene can be converted into two-dimensional materials called graphynes by adding acetylenic connections to a honeycomb structure that contains sp hybridized C-atoms. These structures possess an array of electrical, optical, and mechanical capabilities due to the presence of acetylenic groups within them¹³. Three-dimensional reconstruction and geometric morphology analysis of lunar small craters within the Yutu-2 rover’s patrol range were examined by Xu et al.¹⁴.

Graphdiyne is a novel synthetic carbon-based nanomaterial with sp and $sp^{2}$ hybridized carbon atoms derived from acetylenic groups and benzene rings. This new material has been created by synthesizing $sp^{2}$ and sp-hybridized carbon atoms¹⁵. The $\alpha$, $\beta$, and $\gamma$-type structures are the most notable configurations of graphyne and its derivatives^16,17. Among them, the $\alpha$-type structures’ topological descriptors have been examined^18,19. Because these structures have such vital uses, a research study of the topological descriptors of these different networks would be crucial to compare and contrast the complexities of these structures²⁰. The specular removal of industrial metal Objects without changing lighting configuration was analysed by Chen et al.²¹.

Graphene derivatives, such as graphyne and graphdiyne, have received a lot of attention in mathematics and science because of their outstanding optical, electrical, and mechanical capabilities. These materials have various mathematical applications, including Graphyne and graphdiyne are combinatorial graphs in combinatorics that possess features that can be examined through graph theory. Xu et al.²² used attentive GAN to analyse and remove highlights from a single greyscale image. Graph-theoretic concepts including graph colouring, graph embedding, and graph isomorphism are studied about these materials. Graphyne and graphdiyne are used to analyze the behaviour of different topological descriptors since they are numerical invariants that provide information on the topology of a molecular graph.

The graphene derivatives are helpful in quantum mechanics to examine how electrons behave in a two-dimensional lattice structure. Quantum mechanics can be applied to investigate the electrical characteristics of these materials. Tight-binding approximation (TBA) and density functional theory (DFT) are two methodologies for predicting these materials’ electronic structures²³.

These materials are significant for further research and development because graphene derivatives, such as graphyne and graphdiyne, have a wide range of mathematical applications across several fields. Physical attributes of both structures are determined by their topological descriptors^24,25. It has been found that adding metallic elements and metallic binding to these carbonaceous materials significantly increases the compressive energy of the metallic element. These substances can bind molecules with substantially reduced sorption energies because of their almost uniform metallic atom charges²⁶. The estimated maximum sorption enthalpy for near-room temperature element storage (3.6 kcal/mol) roughly matches the crucial computed enthalpies for element sorption, which vary from 3.5 to 2.8 kcal/mol. Planar carbon allotropes with tunable bandgaps made possible by changing the quantity of alphabetically connected bridging units show promise for use in microelectronics. The cycle-consistent generative adversarial network-based nighttime road scene image enhancement was discussed by Jia et al.²⁷.

A molecular graph is a graphic representation of a chemical compound’s structural formula in which the vertices, or nodes, stand in for atoms and the edges indicate bonds between atoms^28,29. If $\mathcal {G}$ is a graph, then the fundamental symbols and definitions that are utilized, like $d_\lambda$, which stands for the degree of the vertex $\lambda$, are taken from the book that is mentioned in³⁰. Topological descriptors ($\mathbb{T}\mathbb{D}$) are used to determine the graphical structures of chemical compounds, and graph invariants could be used. $\mathbb{T}\mathbb{D}$ are fundamentally represented by converting a chemical graph to a numerical value. Wiener suggests to use $\mathbb{T}\mathbb{D}$ in 1947. He originally reported this index (W) on trees and examined how it was used to correlate the physical features of alcohols, alkanes, and related complexes^31,32.

Topological descriptors, which are structural invariants grounded in molecular graphs and which capture the fundamental connectivity of molecular networks, have drawn a lot of interest lately due to their applications in the fields of quantitative structure-activity and quantitative structure-property relations (QSPR) relations^33,34.

The predictive potential of distance-based and spectrum-based topological descriptors for measuring the $\pi$ electron energy of benzenoid hydrocarbons was discussed by Hayat et al.^35,36,37,38 and Malik et al.³⁹, with applications to carbon nanotubes and boron $\alpha$- and triangular-nanotubes, respectively. Cheminformatics is an emerging discipline that supports QSPRs, which are frequently employed to forecast the bioactivities and characteristics of chemical compounds^40,41,42,43.

In QSPR research, topological descriptors combined with entropy measures may be a more effective tool. Quality tests of spectrum-based valency and distance-based molecular descriptors for polycyclic aromatic and benzenoid hydrocarbons with applications to carbon nanotubes and nanocones were conducted by Hayat et al.^44,45,46,47.

Physicochemical and topological descriptors have been utilized to predict the bioactivity of organic compounds^48,49,50. In a chemical graph, atoms or compounds are represented by the vertices, and their chemical interactions are represented by the contacts. Recently, Eryaşar et al.⁵¹ introduced new formulas and new bounds for the First and Second Zagreb descriptors of Phenylenes. Öztürk Sözen et al.^53,54 investigated an algebraic approach to calculate some topological descriptors and QSPR analysis of some novel drugs used in the treatment of breast cancer and COVID-19. Twisted relative rota-Baxter operators on Leibniz conformal algebras were studied by Guo et al.⁵².

Topological descriptors employing Reduced Reverse Degree $\mathcal {M}$-polynomial have not been investigated for graphenes’ $\alpha$ and $\beta$ structures.

$\mathbb{T}\mathbb{D}$ delineates the graph’s structure, while numerical graph invariants. According to West et al.⁵⁵, the degree in any vertex is represented with $d_\lambda$ or $d(\lambda )$ and represents the number of edges that intersect that vertex $\lambda$. Many researchers are currently performing QSPR investigations of various molecules because it is a more economical way to test compounds than evaluating them^56,57.

In this article, we have provided results for the computation of reduced reverse degree-based topological descriptors ($\mathbb{T}\mathbb{D}$) for graphyne and graphdiyne. Notably, our work introduced a novel approach for analyzing $\mathbb{T}\mathbb{D}$ which is Reduced Reverse Degree $\mathcal {M}$-polynomial builds upon the foundation laid by Zaman et.al⁵⁸.

Objectives of the study

This study’s primary goal is to provide the reduced reverse degree-based graph polynomial and integral and differential operators. Using these operators, the formulation of topological descriptors can be determined. Our goal is to calculate the physical and chemical properties of certain Molecular Graphs using this methodology.

Novelity in the study

We present a novel notion called the reduced reverse degree-based graph polynomial. Polynomial differential and integral operators are formulated based on this. This enables us to create topological descriptors depending on the reduced reverse degree. To execute our methodology, we obtained molecular graphs of $\alpha$-Graphyne, $\beta$-Graphyne, and $\alpha$-Graphdiyne, as depicted in Figs. 1, 2 and 3. We then assessed the physicochemical attributes of these Molecular Graphs using the data computed by our methodology.

Material and methodolgy

Structure of graphyne and graphdiyne

Graphene’s related graphyne and graphdiyne are carbon-based materials with different carbon atom arrangements giving them different structures and physical characteristics. Graphene and graphyne are two-dimensional lattices made of carbon atoms, but graphyne has extra triple bonds connecting some of the carbon atoms. A variation of graphyne known as graphdiyne has two successive triple bonds between some carbon atoms in its structure. Like graphene, both graphyne and graphdiyne are carbon derivatives, but they differ in their bonding arrangements, giving them distinct characteristics. Graphene and graphdiyne belong to the family of carbon-based materials that provide different bonding patterns to increase the versatility of graphene and give it unique qualities.

One carbon allotrope is graphyne. Its structure consists of a planar sheet of sp and $sp^{2}$ linked carbon atoms organized in a crystal lattice, one atom thick. Graphyne is a graphene derivative where the hexagons are connected by acetylenic bonds, as seen in Figs. 1 and 3. In 1987, Baughman et al.⁵⁹ made the initial proposal for graphyne as part of a larger study into the characteristics of novel forms of carbon that had been reported occasionally but not thoroughly examined. Graphyne’s unique electrical structure distinguishes it from other carbon-based materials like diamond and graphite. The most common form of graphyne is $\alpha$-Graphyne and $\beta$-Graphyne.

Significant scientific effort has been directed towards other two-dimensional materials following the discovery of graphene and the prediction of graphyne. Graphdiyne is one among those, it is a variation of graphyne that has two acetylenic links in each unit cell instead of graphyne’s single bond, as shown in Fig. 2. The acetylenic links double the length of the carbon chains that connect the hexagonal rings. In 1997, Haley et al.⁶⁰ made the first prediction about graphdiyne. Initially, the material was created by synthesizing it from related organic molecules and estimating its qualities using computational simulations of associated materials. Graphdiyne belongs to the graphyne family, but because of its distinctive features, it is usually treated as an independent entity^61,62.

Computational techniques

A connected graph $\mathcal {G}$ with vertex and edge sets $\mathcal {V(G)}$ and $\mathcal {E(G)}$, respectively, can be used to simulate a chemical structure⁶³. The number of edges of $\mathcal {G}$ incident with vertex $\lambda$ is called the degree of vertex $\lambda$. The idea of reverse degree vertex $\mathcal {R_D}(\lambda )$, introduced by Kulli⁶⁴, is defined as follows:

$$\begin{aligned} \mathcal {R_D}(\lambda )= \delta (\mathcal {G})-d(\lambda )+1 \end{aligned}$$

Inspiring by this concept Ravi⁶⁵ defines the reduced reverse degree as:

$$\begin{aligned} \mathcal {RR_D}(\lambda )= \delta (\mathcal {G})-d(\lambda )+2 \end{aligned}$$

The degree and reduced reverse degree of an atom is denoted by $d(\lambda )$ and $\mathcal {RR_D}(\lambda )$ of the vertex $\lambda \in \mathcal {V(G)}$ respectively, whereas the maximum degree and maximum reduced reverse degree over all the vertices of $\mathcal {G}$ is denoted by $\delta (\mathcal {G})$ and $\chi (\mathcal {G})$ respectively. Consider the set $\mathcal {RR_D} = \{(i,j\in \mathbb {N}X\mathbb {N}):1\le i \le j \le \chi \}$. We denote $\mathcal {RR_D}({i,j}) = \{\lambda \tau \in \mathcal {E}(\mathcal {G}):\ \mathcal {RR_D}(\lambda )=i,\ \mathcal {RR_D}(\tau )=j \}$. The modified reverse degree⁶⁶ is defined as

$$\begin{aligned} M_k(\mathcal {R_D})= {\left\{ \begin{array}{ll} \delta (\mathcal {G})-d(\lambda )+k: & k\le d(\lambda )\\ \delta (\mathcal {G})-d(\lambda )+k (mod(\delta (\mathcal {G})): & k > d(\lambda ) \end{array}\right. } \end{aligned}$$

The $\mathcal {M}$-Polynomial⁶⁷ is defined as follows:

$$\begin{aligned} \mathcal {M}(G;k,q)= \sum \limits _{i\le j} \rho _{(i,j)} {k^i}{q^j} \end{aligned}$$

(1)

where $\rho _{(i,j)}$ denotes the number of edges in the graph $\mathcal {G}$ for any pair of indices i and j where $i\le j$.

In this study, we have initiated the Reduced reverse degree based $\mathcal {RR_D\,M}$-Polynomial, defined as follows in consensus with the previous studies of $\mathcal {M}$-Polynomial⁶⁸.

$$\begin{aligned} \mathcal {RR_D\,M}(\mathcal {G};l,m)= \sum \limits _{i\le j} \nu {(i,j)} {l^i}{m^j} \end{aligned}$$

(2)

$\nu {(i,j)}$ is counted as number of edges $\lambda \tau \in \mathcal {E}(\mathcal {G})$ such that $\{\mathcal {RR_D}(\lambda ),\mathcal {RR_D}(\tau )\}=\{i,j\}$.

Table 1 contains the formulas that connect the reduced reverse degree-based topological descriptors to the $\mathcal {RR_D\,M}$-Polynomial. In Table 1, operators are defined as below:

$$\begin{aligned} \mathcal {D}_l= & l\frac{\partial ( \mathcal {RR_D\,M}(\mathcal {G};l,m))}{\partial l} \end{aligned}$$

(3)

$$\begin{aligned} \mathcal {D}_m= & m\frac{\partial ( \mathcal {RR_D\,M}(\mathcal {G};l,m))}{\partial m} \end{aligned}$$

(4)

$$\begin{aligned} \mathcal {I}_{l}= & \int _{0}^{l}\frac{1}{z} {(\mathcal {RR_D\,M}(\mathcal {G};z,m))}dz \end{aligned}$$

(5)

$$\begin{aligned} \mathcal {I}_{m}= & \int _{0}^{m}\frac{1}{z} {(M(\mathcal {G};l, z))}dz \end{aligned}$$

(6)

$$\begin{aligned} J(g(l,m))= & g(l,l) \end{aligned}$$

(7)

$$\begin{aligned} Q_\alpha= & l^{\alpha }g(l,m \end{aligned}$$

(8)

Table 1 Topological descriptors derivation formula from $\mathcal {RR_D\,M}$-polynomial.

Results

This section contains the findings of the study. This section begins with the formulation of analytical formulas for a variety of reduced reverse degree-based topological descriptors of $\alpha$ and $\beta$ types for graphene, graphyne, and graphdiyne nanoribbons. Next, using the derived topological descriptors, probabilistic numerical values for these nanoribbons are computed and tabulated. The resulting numerical values are then graphically shown using QSPR modeling of various structures with various topological descriptors and physiochemical features. Figures 1, 2 and 3 displays the structures of the different graphene derivatives that we are interested in. We shall use the notaions ${\mathbb{G}\mathbb{Y}}_{\alpha }$ and ${\mathbb{G}\mathbb{Y}}_{\beta }$ to denote $\alpha$-graphyne and $\beta$ graphyne, ${\mathbb{G}\mathbb{D}}_{\alpha }$ to denote $\alpha$-graphdiyne structures.

Reduce reverse degree based $\boldsymbol{\mathcal {RR_D\,M}}$-polynomials for α-graphyne

Using Fig. 1, the reduced reverse degree-based edge division of $\alpha$-graphyne is described below in Table 2.

Figure 1

Structure of $\alpha$-graphyne.

Table 2 Reduce reverse degree based edge partition of $\alpha$-graphyne.

The Reverse Degree-Based$\mathcal {M}$-Polynomial for$\mathbb {GY_{\alpha }}(r,t)$

Let ${\mathbb{G}\mathbb{Y}}_{\alpha }(r,t)$ be graph of $\alpha$-graphyne. Let $\mathcal {RR_D}({i,j})$ be the set of all edges with a reduced reverse degree of end vertices i, j and $\nu _{i,j}$ be the number of edges in $\mathcal {RR_D}({i,j})$.

$$\begin{aligned} \mathcal {RR_D}({i,j}) & = \{\lambda \tau \in \mathcal {E}(\mathcal {G}):\ \mathcal {RR_D}(\lambda )=i,\ \mathcal {RR_D}(\tau )=j \}\\ \mathcal {RR_D}({3,3}) & = \{\lambda \tau \in \mathcal {E}(\mathcal {G}):\ \mathcal {RR_D}(\lambda )=3,\ \mathcal {RR_D}(\tau )=3 \}\\ \mathcal {RR_D}({3,2}) & = \{\lambda \tau \in \mathcal {E}(\mathcal {G}):\ \mathcal {RR_D}(\lambda )=3,\ \mathcal {RR_D}(\tau )=2 \} \end{aligned}$$

From Fig. 1 and Table 2, it is clear that

$$\begin{aligned} \nu _{(3,3)} = 6rt+3r+9t,\quad \nu _{(3,2)} = 12rt-6r-6t \end{aligned}$$

The $\mathcal {RR_D\,M}$-Polynomial of ${\mathbb{G}\mathbb{Y}}_{\alpha }(r,t)$ is obtained as follow

$$\begin{aligned} \mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha })= & \sum \limits _{i\le j} \nu _{(i,j)}{l^i}{m^j}\\= & \nu _{(3,3)}{l^3}{m^3}+\nu _{(3,2)}{l^3}{m^2} \end{aligned}$$

Putting values of $\nu _{(i,j)}$, we obtain

$$\begin{aligned} {\mathcal{R}\mathcal{R}}_{\mathcal {D}}\,\mathcal {M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m) =(6rt+3r+9t){l^3}{m^3}+(12rt-6r-6t){l^3}{m^2} \end{aligned}$$

(9)

• The differential operators for${\mathbb{G}\mathbb{Y}}_{\alpha }(r,t)$

In view of Table 1, using operators (3) and (4) along with Eq. (9), we get

$$\begin{aligned} \mathcal {D}_l{(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))}= & (18rt+9r+27t){l^3}{m^3}+(36rt-18r-18t){l^3}{m^2} \end{aligned}$$

(10)

$$\begin{aligned} \mathcal {D}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (18rt+9r+27t){l^3}{m^3}+(24rt-12r-12t){l^3}{m^2} \end{aligned}$$

(11)

• The integral operators for${\mathbb{G}\mathbb{Y}}_{\alpha }(r,t)$

In view of Table 1, using operators (5) and (6) along with Eq. (9), we get

$$\begin{aligned} & \mathcal {I}_l(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= (2rt+r+3t){l^3}{m^3}+(4rt-2r-2t){l^3}{m^2} \end{aligned}$$

(12)

$$\begin{aligned} & \mathcal {I}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= (2rt+r+3t){l^3}{m^3}+(6rt-3r-3t){l^3}{m^2} \end{aligned}$$

(13)

• Topological descriptors of$\alpha$-Graphyne using$\mathcal {RR_D\,M}$-Polynomial approach

Reduced Reverse 1st Zagreb Index of${\mathbb{G}\mathbb{Y}}_{\alpha }(r,t)$

In view of Table 1, adding Eqs. (10) and (11), we get

$$\begin{aligned} (\mathcal {D}_l+\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (18rt+9r+27t){l^3}{m^3}+(36rt-18r-18t){l^3}{m^2}\\ & +(18rt+9r+27t){l^3}{m^3}\\ & +(24rt-12r-12t){l^3}{m^2} \\ (\mathcal {D}_l+\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))\big |_{l=m=1}= & [(36rt+18r+54t){l^3}{m^3}+(60rt-30r-30t){l^3}{m^2}]\big |_{l=m=1} \\ \mathcal {RR_D}M_1({\mathbb{G}\mathbb{Y}}_{\alpha })= & 96rt-12r+24t \end{aligned}$$

Reduced Reverse 2nd Zagreb Index of${\mathbb{G}\mathbb{Y}}_{\alpha }(r,t)$

In view of Table 1, applying differential operator on Eq. (11), we have

$$\begin{aligned} \mathcal {D}_l[(\mathcal {D}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))]= & \mathcal {D}_l[(18rt+9r+27t){l^3}{m^3}+(24rt-12r-12t){l^3}{m^2}]\\ (\mathcal {D}_l\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))\big |_{l=m=1}= & [(54rt+27r+81t){l^3}{m^3}+(72rt-36r-36t){l^3}{m^2}]\big |_{l=m=1}\\ \mathcal {RR_D} M_2({\mathbb{G}\mathbb{Y}}_{\alpha })= & 126rt-9r+45t \\ \end{aligned}$$

Reduced Reverse Forgotten Index of${\mathbb{G}\mathbb{Y}}_{\alpha }(r,t)$

In view of Table 1, applying differential operators on Eqs. (10) and (11), we have

$$\begin{aligned} \mathcal {D}^2_l(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (54rt+27r+81t){l^3}{m^3}+(108rt-54r-54t){l^3}{m^2} \\ \mathcal {D}^2_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (54rt+27r+81t){l^3}{m^3}+(48rt-24r-24t){l^3}{m^2} \\ (\mathcal {D}^2_l+\mathcal {D}^2_m)(M(G {\mathbb{G}\mathbb{Y}}_\alpha ;x,y))= & (54rt+27r+81t){l^3}{m^3}+(108rt-54r-54t){l^3}{m^2}\\ & +(54rt+27r+81t){l^3}{m^3}\\ & +(48rt-24r-24t){l^3}{m^2} \\ (\mathcal {D}^2_l+\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (108rt+54r+162t){l^3}{m^3}+(156rt-78r-78t){l^3}{m^2}\\ (\mathcal {D}^2_l+\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))\big |_{l=m=1}= & (108rt+54r+162t){l^3}{m^3}+(156rt-78r-78t){l^3}{m^2}\big |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}F({\mathbb{G}\mathbb{Y}}_{\alpha })= & 264rt-24r+84t \end{aligned}$$

Reduced Reverse Hyper 1st Zagreb Index of${\mathbb{G}\mathbb{Y}}_{\alpha }(r,t)$

In the view of Table 1, and after applying differential operator on Eqs. (10) and (11), we have

$$\begin{aligned} (\mathcal {D}^2_l+\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (54rt+27r+81t){l^3}{m^3}+(108rt-54r-54t){l^3}{m^2}+(54rt+27r+81t){l^3}{m^3}\\ & +(48rt-24r-24t){l^3}{m^2}\\ (2\mathcal {D}_l\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (108rt+54r+162t){l^3}{m^3}+(144rt-72r-72t){l^3}{m^2}\\ (\mathcal {D}_l+\mathcal {D}_m)^2({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))\big |_{l=m=1}= & (216rt+108r+324t){l^3}{m^3}+(300rt-150r-150t){l^3}{m^2}\big |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}HM_1({\mathbb{G}\mathbb{Y}}_{\alpha })= & 516rt-42r+174t \end{aligned}$$

Reduced Hyper 2nd Zagreb Index of${\mathbb{G}\mathbb{Y}}_{\alpha }(r,t)$

In the view of Table 1 after applying differential operator on Eq. (11), we have

$$\begin{aligned} (\mathcal {D}^2_l\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & \mathcal {D}^2_l[(54rt+27r+81t){l^3}{m^3}+(48rt-24r-24t){l^3}{m^2}]\\ (\mathcal {D}^2_l\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))\big |_{l=m=1}= & (486rt+243r+729t){l^3}{m^3}+(432rt-216r-216t){l^3}{m^2}\big |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}HM_{2}({\mathbb{G}\mathbb{Y}}_{\alpha })= & 918rt+27r+513t \end{aligned}$$

Reduced Reverse Sigma Index of${\mathbb{G}\mathbb{Y}}_{\alpha }(r,t)$

In the view Table 1, and after applying differential operator on Eqs. (10) and (11), we have

$$\begin{aligned} (\mathcal {D}^2_l+\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & 54rt+27r+81t){l^3}{m^3}+(108rt-54r-54t){l^3}{m^2}\\ & +(54rt+27r+81t){l^3}{m^3}\\ & +(48rt-24r-24t){l^3}{m^2}\\ (2\mathcal {D}_l\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (108rt+54r+162t){l^3}{m^3}+(144rt-72r-72t){l^3}{m^2}\\ (\mathcal {D}_l-\mathcal {D}_m)^2(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))\big |_{l=m=1}= & (12rt-6r-6t){l^3}{m^2}\big |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}\,\sigma ({\mathbb{G}\mathbb{Y}}_{\alpha })= & 12rt-6r-6t \end{aligned}$$

Reduced Reverse Second Modified Zagreb Index of${\mathbb{G}\mathbb{Y}}_{\alpha }(r,t)$

From Table 1, Second Modified Zagreb Index for $\mathcal {RR_D\,M}$-Polynomial is

$$\begin{aligned} (\mathcal {I}_l\mathcal {I}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & \mathcal {I}_l[\mathcal {I}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m)]\\= & \mathcal {I}_l[(2rt+r+3t){l^3}{m^3}+(6rt-3r-3t){l^3}{m^2} ] \end{aligned}$$

After operating integral operator $\mathcal {I}_l$ (5), we have

$$\begin{aligned} (\mathcal {I}_l\mathcal {I}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))\big |_{l=m=1}= & \left[ \left( \frac{2}{3}rt+\frac{1}{3}r+t\right) {l^3}{m^3}+(2rt-r-t){l^3}{m^2}\right] \bigg |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}\,^mM_2({\mathbb{G}\mathbb{Y}}_{\alpha })= & \frac{8}{3}rt-\frac{2}{3}r \end{aligned}$$

Reduced Reverse Redefined Third Zagreb Index

In view of Table 1, Reduced Reverse Redefined Third Zagreb Index for $\mathcal {RR_D\,M}$-Polynomial is computed as follow

$$\begin{aligned} (\mathcal {D}_l+\mathcal {D}_m)(\mathcal {M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (36rt+18r+54t){l^3}{m^3}+(60rt-30r-30t){l^3}{m^2} \\ \mathcal {D}_m(\mathcal {D}_l+\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m)= & (108rt+54r+162t){l^3}{m^3}+(120rt-60r-60t){l^3}{m^2} \\ (\mathcal {D}_l\mathcal {D}_m)(\mathcal {D}_l+\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (324rt+162r+486t){l^3}{m^3}+(360rt-180r-180t){l^3}{m^2}\\ (\mathcal {D}_l\mathcal {D}_m)(\mathcal {D}_l+\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))\big |_{l=m=1}= & (324rt+162r+486t){l^3}{m^3}+(360rt-180r-180t){l^3}{m^2})\big |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}\, ReZG_3({\mathbb{G}\mathbb{Y}}_{\alpha };r,t)= & 684rt-18r+306t\\ \end{aligned}$$

Reduced Reverse Symmetric Division Degree Index of${\mathbb{G}\mathbb{Y}}_{\alpha }(r,t)$

In view Table 1 using Eq. (13), we have

$$\begin{aligned} (\mathcal {I}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (2rt+r+3t){l^3}{m^3}+(6rt-3r-3t){l^3}{m^2} \end{aligned}$$

After applying differential operator $\mathcal {D}_l$ (3), we have

$$\begin{aligned} (\mathcal {D}_l\mathcal {I}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (6rt+3r+9t){l^3}{m^3}+(18rt-9r-9t){l^3}{m^2} \end{aligned}$$

Also from Eq. (11)

$$\begin{aligned} \mathcal {D}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (18rt+9r+27t){l^3}{m^3}+(24rt-12r-12t){l^3}{m^2} \end{aligned}$$

After operating integral operator $\mathcal {I}_l$ (5), we have

$$\begin{aligned} \mathcal {I}_l\mathcal {D}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (6rt+3r+9t){l^3}{m^3}+(8rt-4r-4t){l^3}{m^2}\\ (\mathcal {D}_l\mathcal {I}_m+\mathcal {I}_l\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (12rt+6r+18t){l^3}{m^3}+(26rt-13r-13t){l^3}{m^2}\\ (\mathcal {D}_l\mathcal {I}_m+\mathcal {I}_l\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))\big |_{l=m=1}= & (12rt+6r+18t){l^3}{m^3}+(26rt-13r-13t){l^3}{m^2}\big |_{l=m=1}\\ \mathcal {RR_D}\,SDD({\mathbb{G}\mathbb{Y}}_{\alpha })= & 48rt-7r+5t \end{aligned}$$

Reduced Reverse Harmonic Index of${\mathbb{G}\mathbb{Y}}_{\alpha }(r,t)$

In view of Table 1, applying operator (8) on Eq. (9), we have

$$\begin{aligned} {J}\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m)= & (6rt+3r+9t){l^6}+(12rt-6r-6t){l^5}\\ \end{aligned}$$

After operating integral operator $\mathcal {I}_l$ (5), we have

$$\begin{aligned} \mathcal {I}_l{J}\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m)= & \left( rt+\frac{1}{2}r+\frac{3}{2}t\right) {l^6}+\left( \frac{12}{5}rt-\frac{6}{5}r-\frac{6}{5}t\right) {l^5}\\ {2\mathcal {I}_lJ}\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m)= & (2rt+r+3t){l^6}+\left( \frac{24}{5}rt-\frac{12}{5}r-\frac{12}{5}t\right) {l^5}\\ {2\mathcal {I}_lJ}\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m)\big |_{l=m=1}= & \left[ (2rt+r+3t){l^6}+\left( \frac{24}{5}rt-\frac{12}{5}r-\frac{12}{5}t\right) {l^5}\right] \bigg |_{l=1}\\ \end{aligned}$$

$$\begin{aligned} \mathcal {RR_D}\,H({\mathbb{G}\mathbb{Y}}_{\alpha })= & \frac{34}{5}rt-\frac{7}{5}r+\frac{3}{5}t \end{aligned}$$

Reduced Reverse Inverse Sum Index of${\mathbb{G}\mathbb{Y}}_{\alpha }(r,t)$

From Table 1, Harmonic Index for $\mathcal {RR_D\,M}$-Polynomial is

$$\begin{aligned} {\mathcal {D}_l}{\mathcal {D}_m}(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (54rt+27r+81t){l^3}{m^3}+(72rt-36r-36t){l^3}{m^2}\\ {J}{\mathcal {D}_l}{\mathcal {D}_m}(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (54rt+27r+81t){l^6}+(72rt-36r-36t){l^5}\\ \end{aligned}$$

After operating integral operator $\mathcal {I}_l$ (5), we have

$$\begin{aligned} {\mathcal {I}_l}{J}{\mathcal {D}_l}{\mathcal {D}_m}(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & \left( 9rt+\frac{9}{2}r+\frac{27}{2}t\right) {l^6}+\left( \frac{72}{5}rt-\frac{36}{5}r-\frac{36}{5}t\right) {l^5}\\ {\mathcal {I}_l}{J}{\mathcal {D}_l}{\mathcal {D}_m}(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))\big |_{l=1}= & \left( 9rt+\frac{9}{2}r+\frac{27}{2}t\right) {l^6}+\left( \frac{72}{5}rt-\frac{36}{5}r-\frac{36}{5}t\right) {l^5}\big |_{l=1}\\ \end{aligned}$$

$$\begin{aligned} \mathcal {RR_D}\,I({\mathbb{G}\mathbb{Y}}_{\alpha })= & \frac{117}{5}rt-\frac{27}{10}r+\frac{63}{10}t \end{aligned}$$

Reduced Reverse Augmented Index of${\mathbb{G}\mathbb{Y}}_{\alpha }(r,t)$

In view Table 1 applying operator $\mathcal {D}^3_m$ on Eq. (9), we have

$$\begin{aligned} \mathcal {D}^2_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (54rt+27r+81t){l^3}{m^3}+(48rt-24r-24t){l^3}{m^2}\\ \mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (162rt+81r+243t){l^3}{m^3}+(96rt-48r-48t){l^3}{m^2} \end{aligned}$$

After operating differential operator $\mathcal {D}^3_l$, we have

$$\begin{aligned} \mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (4374rt+2187r+6561t){l^3}{m^3}+(2592rt-1296r-1296t){l^3}{m^2} \\ {J}\mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (4374rt+2187r+6561t){l^6}+(2592rt-1296r-1296t){l^5}\\ {Q_{-2}}{J}\mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & {l^{-2}}[(4374rt+2187r+6561t){l^6}+(2592rt-1296r-1296t){l^5}]\\ {Q_{-2}}{J}\mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (4374rt+2187r+6561t){l^4}+(2592rt-1296r-1296t){l^3} \end{aligned}$$

After operating Integral operator $I^3_l$, we have

$$\begin{aligned} {\mathcal {I}^3_l}{Q_{-2}}{J}\mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & \left( \frac{2187}{32}rt+\frac{2187}{64}r+\frac{6561}{64}t\right) {l^4}+(96rt-48r-48t){l^3}\\ {\mathcal {I}^3_l}{Q_{-2}}{J}\mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))\big |_{l=1}= & \bigg [\frac{2187}{32}rt+\frac{2187}{64}r+\frac{6561}{64}t\bigg ){l^4}+(96rt-48r-48t){l^3}\bigg ]\bigg |_{l=1}\\ \mathcal {RR_D}\,A({\mathbb{G}\mathbb{Y}}_{\alpha })= & \frac{5259}{32}rt-\frac{885}{64}r+\frac{3489}{64}t \end{aligned}$$

Reduced reverse degree based $\boldsymbol{\mathcal {RR_{D} M}}$-polynomials for α-graphdiyne

Using Fig. 2, the reduced reverse degree-based edge division of $\alpha$-graphdiyne is described below in Table 3.

Figure 2

Structure of $\alpha$-graphdiyne.

Table 3 Reduced reverse degree based edge partition of $\alpha$-graphdiyne.

Reduce Reverse Degree-Based$\mathcal {M}$-Polynomial for${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$

Let ${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$ be graph of $\alpha$-graphdiyne . Let $\mathcal {RR_D}({i,j})$ be the set of all edges with a reduced reverse degree of end vertices i, j and $\nu _{i,j}$ be the number of edges in $\mathcal {RR_D}({i,j})$.

$$\begin{aligned} \mathcal {RR_D}({i,j}) & = \{\lambda \tau \in \mathcal {E}(\mathcal {G}):\ \mathcal {RR_D}(\lambda )=i,\ \mathcal {RR_D}(\tau )=j \}\\ \mathcal {RR_D}({3,3}) & = \{\lambda \tau \in \mathcal {E}(\mathcal {G}):\ \mathcal {RR_D}(\lambda )=3,\ \mathcal {RR_D}(\tau )=3 \}\\ \mathcal {RR_D}({3,2}) & = \{\lambda \tau \in \mathcal {E}(\mathcal {G}):\ \mathcal {RR_D}(\lambda )=3,\ \mathcal {RR_D}(\tau )=2 \} \end{aligned}$$

From Fig. 2 and Table 3, it is clear that $\nu _{(3,3)}$ = 18rt+r+11t, $\nu _{(3,2)}$ = 12rt-6r-6t

The $\mathcal {RR_D\,M}$-Polynomial of ${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$ is obtained as follow

$$\begin{aligned} \mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha })= & \sum \limits _{i\le j} \nu _{(i,j)}{l^i}{m^j}\\= & \nu _{(3,3)}{l^3}{m^3}+\nu _{(3,2)}{l^3}{m^2} \end{aligned}$$

Putting values of $\nu _{(i,j)}$, we obtain

$$\begin{aligned} \mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m) =(18rt+r+11t){l^3}{m^3}+(12rt-6r-6t){l^3}{m^2} \end{aligned}$$

(14)

• The differential operators for${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$

In view of Table 1, using operators (3) and (4) along with Eq. (14), we get

$$\begin{aligned} \mathcal {D}_l{(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))}= & (54rt+3r+33t){l^3}{m^3}+(36rt-18r-18t){l^3}{m^2} \end{aligned}$$

(15)

and

$$\begin{aligned} \mathcal {D}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (54rt+3r+33t){l^3}{m^3}+(24rt-12r-12t){l^3}{m^2} \end{aligned}$$

(16)

• The integral operators for${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$

In view of Table 1, using operators (5) and (6) along with Eq. (14), we get

$$\begin{aligned} & \mathcal {I}_l(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= \left( 6rt+\frac{1}{3}r+\frac{11}{3}t\right) {l^3}{m^3}+(4rt-3r-3t){l^3}{m^2} \end{aligned}$$

(17)

$$\begin{aligned} & \mathcal {I}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= \left( 6rt+\frac{1}{3}r+\frac{11}{3}t\right) {l^3}{m^3}+(6rt-3r-3t){l^3}{m^2} \end{aligned}$$

(18)

• Topological descriptors of$\alpha$-graphdiyne using$\mathcal {RR_D\,M}$-Polynomial approach

Reduced Reverse 1st Zagreb Index of${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$

In view of Table 1, adding Eqs. (15) and (16), we get

$$\begin{aligned} (\mathcal {D}_l+\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (54rt+3r+33t){l^3}{m^3}+(36rt-18r-18t){l^3}{m^2}\\ & +(54rt+3r+33t){l^3}{m^3}\\ & +(24rt-12r-12t){l^3}{m^2} \\ (\mathcal {D}_l+\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))\big |_{l=m=1}= & [(108rt+6r+66t){l^3}{m^3}+(60rt-30r-30t){l^3}{m^2}]\big |_{l=m=1} \\ \mathcal {RR_D}M_1({\mathbb{G}\mathbb{D}}_{\alpha })= & 168rt-24r+36t \end{aligned}$$

Reduced Reverse 2nd Zagreb Index of${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$

In view of Table 1, applying differential operator on Eq. (16), we have

$$\begin{aligned} \mathcal {D}_l[(\mathcal {D}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))]= & \mathcal {D}_l[(54rt+3r+33t){l^3}{m^3}+(24rt-12r-12t){l^3}{m^2}]\\ (\mathcal {D}_l\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))\big |_{l=m=1}= & [(162rt+9r+99t){l^3}{m^3}+(72rt-36r-36t){l^3}{m^2}]\big |_{l=m=1}\\ \mathcal {RR_D} M_2({\mathbb{G}\mathbb{D}}_{\alpha })= & 234rt-27r+63t \end{aligned}$$

Reduced Reverse Forgotten Index of${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$

In view of Table 1, applying differential operators on Eqs. (15) and (16), we have

$$\begin{aligned} \mathcal {D}^2_l(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (162rt+9r+99t){l^3}{m^3}+(108rt-54r-54t){l^3}{m^2} \\ \mathcal {D}^2_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (162rt+9r+99t){l^3}{m^3}+(48rt-24r-24t){l^3}{m^2} \\ (\mathcal {D}^2_l+\mathcal {D}^2_m)(M(G {\mathbb{G}\mathbb{D}}_{\alpha };x,y))= & (162rt+9r+99t){l^3}{m^3}+(108rt-54r-54t){l^3}{m^2}\\ & +(162rt+9r+99t){l^3}{m^3}\nonumber \\ & +(48rt-24r-24t){l^3}{m^2} \\ (\mathcal {D}^2_l+\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (324rt+18r+198t){l^3}{m^3}+(156rt-78r-78t){l^3}{m^2}\\ (\mathcal {D}^2_l+\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))\big |_{l=m=1}= & (324rt+18r+198t){l^3}{m^3}+(156rt-78r-78t){l^3}{m^2}\big |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}F({\mathbb{G}\mathbb{D}}_{\alpha })= & 480rt-60r+120t \end{aligned}$$

Reduced Reverse Hyper 1st Zagreb Index of${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$

In the view of Table 1, and after applying differential operator on Eqs. (15) and (16), we have

$$\begin{aligned} (\mathcal {D}^2_l+\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (162rt+9r+99t){l^3}{m^3}+(108rt-54r-54t){l^3}{m^2}\\ & +(162rt+9r+99t){l^3}{m^3}\nonumber \\ & +(48rt-24r-24t){l^3}{m^2}\\ (2\mathcal {D}_l\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (324rt+18r+198t){l^3}{m^3}+(144rt-72r-72t){l^3}{m^2}\\ (\mathcal {D}_l+\mathcal {D}_m)^{2}(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))\big |_{l=m=1}= & (648rt+36r+396t){l^3}{m^3}+(144rt-72r-72t){l^3}{m^2}\big |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}HM_1({\mathbb{G}\mathbb{D}}_{\alpha })= & 948rt-114r+246t \end{aligned}$$

Reduced Reverse Hyper 2nd Zagreb Index of${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$

In the view of Table 1, and after applying differential operator $\mathcal {D}^2_l$ on Eq. (16), we have

$$\begin{aligned} (\mathcal {D}^2_l\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & \mathcal {D}^2_l[(162rt+9r+99t){l^3}{m^3}+(48rt-24r-24t){l^3}{m^2}]\\ (\mathcal {D}^2_l\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))\big |_{l=m=1}= & (1458rt+81r+891t){l^3}{m^3}+(432rt-216r-216t){l^3}{m^2}\big |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}HM_2({\mathbb{G}\mathbb{D}}_{\alpha })= & 1890rt-135r+675t \end{aligned}$$

Reduced Reverse Sigma Index of${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$

In the view Table 1, and after applying differential operator on Eqs. (15) and (16), we have

$$\begin{aligned} (\mathcal {D}^2_l+\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (162rt+9r+99t){l^3}{m^3}+(108rt-54r-54t){l^3}{m^2}\\ & +(162rt+9r+99t){l^3}{m^3}\\ & +(48rt-24r-24t){l^3}{m^2}\\ (2\mathcal {D}_l\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (324rt+18r+198t{l^3}{m^3}+(144rt-72r-72t){l^3}{m^2}\\ (\mathcal {D}_l-\mathcal {D}_m)^2(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))\big |_{l=m=1}= & (12rt-6r-6t){l^3}{m^2}\big |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}\,\sigma ({\mathbb{G}\mathbb{D}}_{\alpha })= & 12rt-6r-6t \end{aligned}$$

Reduced Reverse Second Modified Zagreb Index of${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$

In view Table 1, using Eq. (18)

$$\begin{aligned} (\mathcal {I}_l\mathcal {I}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & \mathcal {I}_l[\mathcal {I}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m)]\\= & \mathcal {I}_l\left[ \left( 6rt+\frac{1}{3}r+\frac{11}{3}t\right) {l^3}{m^3}+(6rt-3r-3t){l^3}{m^2} \right] \end{aligned}$$

After operating Integral operator $\mathcal {I}_l$ (5), we have

$$\begin{aligned} (\mathcal {I}_l\mathcal {I}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))\big |_{l=m=1}= & \left[ \left( 2rt+\frac{1}{9}r+\frac{11}{9}t\right) {l^3}{m^3}+(2rt-r-t){l^3}{m^2}\right] \Bigg |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}\,^mM_2({\mathbb{G}\mathbb{D}}_{\alpha })= & 4rt-\frac{8}{9}r+\frac{2}{9}t \end{aligned}$$

Reduced Reverse Redefined Third Zagreb Index of${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$

In view of Table 1, Reduced Reverse Redefined Third Zagreb Index for $\mathcal {RR_D\,M}$-Polynomial is computed as follow

$$\begin{aligned} (\mathcal {D}_l+\mathcal {D}_m)(\mathcal {M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (108rt+6r+66t){l^3}{m^3}+(60rt-30r-30t){l^3}{m^2} \\ \mathcal {D}_m(\mathcal {D}_l+\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m)= & (324rt+18r+198t){l^3}{m^3}+(120rt-60r-60t){l^3}{m^2} \\ (\mathcal {D}_l\mathcal {D}_m)(\mathcal {D}_l+\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (972rt+54r+594t){l^3}{m^3}+(360rt-180r-180t){l^3}{m^2}\\ (\mathcal {D}_l\mathcal {D}_m)(\mathcal {D}_l+\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))\big |_{l=m=1}= & (972rt+54r+594t){l^3}{m^3}+(360rt-180r-180t){l^3}{m^2})\big |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}\, ReZG_3({\mathbb{G}\mathbb{D}}_{\alpha })= & 1332rt-126r+414t\\ \end{aligned}$$

Reduced Reverse Symmetric Division Degree Index of${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$

In view Table 1 using Eq. (18), we have

$$\begin{aligned} (\mathcal {I}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & \left( 6rt+\frac{1}{3}r+\frac{11}{3}t\right) {l^3}{m^3}+(6rt-3r-3t){l^3}{m^2} \end{aligned}$$

After applying operator $\mathcal {D}_l$ (3), we have

$$\begin{aligned} (\mathcal {D}_l\mathcal {I}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (18rt+r+{11}t){l^3}{m^3}+(18rt-9r-9t){l^3}{m^2} \end{aligned}$$

Also from Eq. (16)

$$\begin{aligned} \mathcal {D}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (54rt+3r+33t){l^3}{m^3}+(24rt-12r-12t){l^3}{m^2} \end{aligned}$$

After operating integral operator $\mathcal {I}_l$ (5), we have

$$\begin{aligned} \mathcal {I}_l\mathcal {D}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (18rt+r+11t){l^3}{m^3}+(8rt-4r-4t){l^3}{m^2}\\ (\mathcal {D}_l\mathcal {I}_m+\mathcal {I}_l\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (36rt+2r+22t){l^3}{m^3}+(26rt-13r-13t){l^3}{m^2}\\ (\mathcal {D}_l\mathcal {I}_m+\mathcal {I}_l\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))\big |_{l=m=1}= & (36rt+2r+22t){l^3}{m^3}+(26rt-13r-13t){l^3}{m^2}\big |_{l=m=1}\\ \mathcal {RR_D}\,SDD({\mathbb{G}\mathbb{D}}_{\alpha })= & 62rt-11r+9t \end{aligned}$$

Reduced Reverse Harmonic Index of${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$

In view of Table 1, applying operator (8) on Eq. (14), we have

$$\begin{aligned} {J}\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m)= & (18rt+r+11t){l^6}+(12rt-6r-6t){l^5} \end{aligned}$$

After operating integral operator $\mathcal {I}_l$ (5), we have

$$\begin{aligned} \mathcal {I}_l{J}\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m)= & \left( 3rt+\frac{1}{6}r+\frac{11}{6}t\right) {l^6}+\left( \frac{12}{5}rt-\frac{6}{5}r-\frac{6}{5}t\right) {l^5}\\ {2\mathcal {I}_lJ}\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m)= & \left( 6rt+\frac{1}{3}r+\frac{11}{3}t\right) {l^6}+\left( \frac{24}{5}rt-\frac{12}{5}r-\frac{12}{5}t\right) {l^5}\\ {2\mathcal {I}_lJ}\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m)\big |_{l=m=1}= & \left[ \left( 6rt+\frac{1}{3}r+\frac{11}{3}t\right) {l^6}+\left( \frac{24}{5}rt-\frac{12}{5}r-\frac{12}{5}t\right) {l^5}\right] \bigg |_{l=1}\\ \mathcal {RR_D}\,H({\mathbb{G}\mathbb{D}}_{\alpha })= & \frac{54}{5}rt-\frac{31}{15}r+\frac{19}{5}t \end{aligned}$$

Reduced Reverse Inverse Sum Index of${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$

From Table 1, Harmonic Index for $\mathcal {RR_D\,M}$-Polynomial is

$$\begin{aligned} {\mathcal {D}_l}{\mathcal {D}_m}(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (162rt+9r+99t){l^3}{m^3}+(72rt-36r-36t){l^3}{m^2}\\ {J}{\mathcal {D}_l}{\mathcal {D}_m}(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (162rt+9r+99t){l^6}+(72rt-36r-36t){l^5} \end{aligned}$$

After operating integral operator $\mathcal {I}_l$ (5), we have

$$\begin{aligned} {\mathcal {I}_l}{J}{\mathcal {D}_l}{\mathcal {D}_m}(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & \left( 27rt+\frac{3}{2}r+\frac{33}{2}t\right) {l^6}+\left( \frac{72}{5}rt-\frac{36}{5}r-\frac{36}{5}t\right) {l^5}\\ {\mathcal {I}_l}{J}{\mathcal {D}_l}{\mathcal {D}_m}(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))\big |_{l=1}= & \left( 27rt+\frac{3}{2}r+\frac{33}{2}t\right) {l^6}+\left( \frac{72}{5}rt-\frac{36}{5}r-\frac{36}{5}t\right) {l^5}\bigg |_{l=1}\\ \mathcal {RR_D}\,I({\mathbb{G}\mathbb{D}}_{\alpha })= & \frac{207}{5}rt-\frac{57}{10}r+\frac{93}{10}t \end{aligned}$$

Reduced Reverse Augmented Index of${\mathbb{G}\mathbb{D}}_{\alpha }(r,t)$

In view Table 1 applying operator $\mathcal {D}^3_m$ on Eq. (9), we have

$$\begin{aligned} \mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (486rt+27r+299t){l^3}{m^3}+(96rt-48r-48t){l^3}{m^2}\\ \end{aligned}$$

After operating differential operator $\mathcal {D}^3_l$, we have

$$\begin{aligned} \mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (13122rt+729r+8073t){l^3}{m^3}+(2592rt-1296r-1296t){l^3}{m^2} \\ {J}\mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (13122rt+729r+8073t){l^6}+(2592rt-1296r-1296t){l^5}\\ {Q_{-2}}{J}\mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & {l^{-2}}[(13122rt+729r+8073t){l^6}+(2592rt-1296r-1296t){l^5}]\\ {Q_{-2}}{J}\mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (13122rt+729r+8073t){l^4}+(2592rt-1296r-1296t){l^3}\\ \end{aligned}$$

After operating Integral operator $I^3_l$, we have

$$\begin{aligned} {\mathcal {I}^3_l}{Q_{-2}}{J}\mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & \left( \frac{6561}{32}rt+\frac{729}{64}r+\frac{8073}{64}t\right) {l^4}+(96rt-48r-48t){l^3}\\ {\mathcal {I}^3_l}{Q_{-2}}{J}\mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))\big |_{l=1}= & \bigg [\frac{6561}{32}rt+\frac{729}{64}r+\frac{8073}{64}t\bigg ){l^4}+(96rt-48r-48t){l^3}\bigg ]\Bigg |_{l=1}\\ \mathcal {RR_D}\,A({\mathbb{G}\mathbb{Y}}_{\alpha })= & \frac{9633}{32}rt-\frac{2343}{64}r+\frac{5001}{64}t \end{aligned}$$

Reduced reverse degree based $\boldsymbol{\mathcal {RR_{D} M}}$-polynomials for β-graphyne

Using Fig. 3, the reduced reverse degree-based edge division of $\alpha$-graphdiyne is described below in Table 4.

Figure 3

Structure of $\beta$-graphyne.

Table 4 Reduced reverse degree based edge partition of $\beta$-graphyne.

The Reduce Reverse Degree-Based$\mathcal {M}$-Polynomial for${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$

Let ${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$ be graph of $\beta$-graphyne. Let $\mathcal {RR_D}({i,j})$ be the set of all edges with a reduced Reverse degree of end vertices i, j and $\nu _{i,j}$ be the number of edges in $\mathcal {RR_D}({i,j})$.

$$\begin{aligned} \mathcal {RR_D}({i,j}) & = \{\lambda \tau \in \mathcal {E}(\mathcal {G}):\ \mathcal {RR_D}(\lambda )=i,\ \mathcal {RR_D}(\tau )=j \}\\ \mathcal {RR_D}({3,3}) & = \{\lambda \tau \in \mathcal {E}(\mathcal {G}):\ \mathcal {RR_D}(\lambda )=3,\ \mathcal {RR_D}(\tau )=3 \}\\ \mathcal {RR_D}({3,2}) & = \{\lambda \tau \in \mathcal {E}(\mathcal {G}):\ \mathcal {RR_D}(\lambda )=3,\ \mathcal {RR_D}(\tau )=2 \}\\ \mathcal {RR_D}({2,2}) & = \{\lambda \tau \in \mathcal {E}(\mathcal {G}):\ \mathcal {RR_D}(\lambda )=2,\ \mathcal {RR_D}(\tau )=2 \} \end{aligned}$$

From Fig. 3 and Table 4, it is clear that

$\nu _{(3,3)}$ = 12rt+6r+18t, $\nu _{(3,2)}$ = 24rt-4r+4t , $\nu _{(2,2)}$ = 6rt-r+t

The $\mathcal {RR_D\,M}$-Polynomial of ${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$ is obtained as follow

$$\begin{aligned} \mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta })= & \sum \limits _{i\le j} \nu _{(i,j)}{l^i}{m^j}\\= & \nu _{(3,3)}{l^3}{m^3}+\nu _{(3,2)}{l^3}{m^2}+\nu _{(2,2)}{l^2}{m^2} \end{aligned}$$

Putting values of $\nu _{(i,j)}$, we obtain

$$\begin{aligned} \mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m) =(12rt+6r+18t){l^3}{m^3}+(24rt-4r+4t){l^3}{m^2}+(6rt-r+t){l^2}{m^2} \end{aligned}$$

(19)

• The differential operators for${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$

In view of Table 1, using operators (3) and (4) along with Eq. (19), we get

$$\begin{aligned} \mathcal {D}_l{(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))}= & (36rt+18r+54t){l^3}{m^3}+(72rt-12r+12t){l^3}{m^2}+(12rt-2r+2t){l^2}{m^2} \end{aligned}$$

(20)

$$\begin{aligned} \mathcal {D}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (36rt+18r+54t){l^3}{m^3}+(48rt-8r+8t){l^3}{m^2}+(12rt-2r+2t){l^2}{m^2} \end{aligned}$$

(21)

• The integral operators for${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$

In view of Table 1, using operators (5) and (6) along with Eq. (19), we get

$$\begin{aligned} \mathcal {I}_l(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (4rt+2r+6t){l^3}{m^3}+\left( 8rt-\frac{4}{3}r+\frac{4}{3}t\right) {l^3}{m^2}+\left( 3rt-\frac{1}{2}r+\frac{1}{2}t\right) {l^2}{m^2} \end{aligned}$$

(22)

$$\begin{aligned} \mathcal {I}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (4rt+2r+6t){l^3}{m^3}+(12rt-2r+2t){l^3}{m^2}+\left( 3rt-\frac{1}{2}r+\frac{1}{2}t\right) {l^2}{m^2} \end{aligned}$$

(23)

• Topological descriptors of$\beta$-Graphyne using$\mathcal {RR_D\,M}$-Polynomial approach

Reduced Reverse 1st Zagreb Index of${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$

In view of Table 1, adding Eqs. (20) and (21), we get

$$\begin{aligned} (\mathcal {D}_l+\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (36rt+18r+54t){l^3}{m^3}+(72rt-12r+12t){l^3}{m^2}\\ & +(12rt-2r+2t){l^2}{m^2}\\ & +(36rt+18r+54t){l^3}{m^3}+(48rt-8r+8t){l^3}{m^2}\\ & +(12rt-2r+2t){l^2}{m^2} \\ (\mathcal {D}_l+\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))\big |_{l=m=1}= & [(72rt+36r+108t){l^3}{m^3}+(120rt-20r+20t){l^3}{m^2}\\ & +(24rt-4r+4t){l^2}{m^2}]\big |_{l=m=1} \\ \mathcal {RR_D}M_1({\mathbb{G}\mathbb{Y}}_{\beta })= & 216rt+12r+132t \end{aligned}$$

Reduced Reverse 2nd Zagreb Index of${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$

In view of Table 1, applying differential operator on Eq. (21), we have

$$\begin{aligned} \mathcal {D}_l[(\mathcal {D}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))]= & \mathcal {D}_l[ (36rt+18r+54t){l^3}{m^3}+(48rt-8r+8t){l^3}{m^2}\\ & +(12rt-2r+2t){l^2}{m^2}]\\ (\mathcal {D}_l\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))\big |_{l=m=1}= & [ (108rt+54r+162t){l^3}{m^3}+(144rt-24r+24t){l^3}{m^2}\\ & +(24rt-4r+4t){l^2}{m^2}]\big |_{l=m=1}\\ \mathcal {RR_D} M_2({\mathbb{G}\mathbb{Y}}_{\beta })= & 276rt+26r+190t \end{aligned}$$

Reduced Reverse Forgotten Index of${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$

In view of Table 1, applying differential operators on Eqs. (20) and (21), we have

$$\begin{aligned} \mathcal {D}^2_l(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (108rt+54r+162t){l^3}{m^3}+(216rt-36r+36t){l^3}{m^2}\\ & +(24rt-4r+4t){l^2}{m^2}\\ \mathcal {D}^2_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (108rt+54r+162t){l^3}{m^3}+(96rt-16r+16t){l^3}{m^2}\\ & +(24rt-4r+4t){l^2}{m^2}\\ (\mathcal {D}^2_l+\mathcal {D}^2_m)(M( (\mathcal {D}^2_l+\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (216rt+108r+324t){l^3}{m^3}+(312rt-52r+52t){l^3}{m^2}\\ & +(48rt-8r+8t){l^2}{m^2}\\ (\mathcal {D}^2_l+\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))\big |_{l=m=1}= & (216rt+108r+324t){l^3}{m^3}+(312rt-52r+52t){l^3}{m^2}\\ & +(48rt-8r+8t){l^2}{m^2}{l^3}{m^2}\big |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}F({\mathbb{G}\mathbb{Y}}_{\beta })= & 576rt+48r+384t \end{aligned}$$

Reduced Reverse Hyper 1st Zagreb Index of${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$

In the view of Table 1, and after applying differential operator on Eqs. (20) and (21), we have

$$\begin{aligned} (\mathcal {D}^2_l+\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (216rt+108r+324t){l^3}{m^3}+(312rt-52r+52t){l^3}{m^2}\\ & +(48rt-8r+8t){l^2}{m^2}\\ (2\mathcal {D}_l\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (216rt+108r+324t){l^3}{m^3}+(288rt-48r+48t){l^3}{m^2}\\ & +(48rt-8r+8t){l^2}{m^2}\\ (\mathcal {D}_l+\mathcal {D}_m)^2({\mathbb{G}\mathbb{Y}}_{\beta };l,m))\big |_{l=m=1}= & (432rt+216r+648t){l^3}{m^3}+(600rt-100r+100t){l^3}{m^2}\\ & +(96rt-16r+16t){l^2}{m^2}\big |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}HM_1({\mathbb{G}\mathbb{Y}}_{\beta })= & 1128rt+100r+764t \end{aligned}$$

Reduced Reverse Hyper 2nd Zagreb Index of${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$

In the view of Table 1, and after applying differential operator on Eq. (21), we have

$$\begin{aligned} (\mathcal {D}^2_l\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & \mathcal {D}^2_l[(108rt+54r+162t){l^3}{m^3}+(96rt-16r+16t){l^3}{m^2}\\ & +(24rt-4r+4t){l^2}{m^2}]\\ (\mathcal {D}^2_l\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))\big |_{l=m=1}= & (972rt+486r+1458t){l^3}{m^3}+(864rt-144r+144t){l^3}{m^2}\\ & +(96rt-16r+16t){l^2}{m^2}\big |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}HM_2({\mathbb{G}\mathbb{Y}}_{\beta })= & 1932rt+326r+1618t \end{aligned}$$

Reduced Reverse Sigma Index of${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$

In the view Table 1, and after applying differential operator on Eqs. (20) and (21), we have

$$\begin{aligned} (\mathcal {D}^2_l+\mathcal {D}^2_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (216rt+108r+324t){l^3}{m^3}+(312rt-52r+52t){l^3}{m^2}\\ & +(48rt-8r+8t){l^2}{m^2}\\ (2\mathcal {D}_l\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (216rt+108r+324t){l^3}{m^{3}}+(288rt-48r+48t){l^3}{m^2}\\ & +(48rt-8r+8t){l^2}{m^2}\\ (\mathcal {D}_l-\mathcal {D}_m)^2(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))\big |_{l=m=1}= & (24rt-4r+4t){l^3}{m^2}\big |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}\,\sigma ({\mathbb{G}\mathbb{Y}}_{\beta })= & 24rt-4r+4t \end{aligned}$$

Reduced Reverse Second Modified Zagreb of${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$

From Table 1, Second Modified Zagreb Index for $\mathcal {RR_D\,M}$-Polynomial is

$$\begin{aligned} (\mathcal {I}_l\mathcal {I}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & \mathcal {I}_l[\mathcal {I}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m)]\\= & \mathcal {I}_l\left[ (4rt+2r+6t){l^3}{m^3}+(12rt-2r+2t){l^3}{m^2}+\left( 3rt-\frac{1}{2}r+\frac{1}{2}t\right) {l^2}{m^2}\right] \end{aligned}$$

After operating integral operator $\mathcal {I}_l$ (5), we have

$$\begin{aligned} (\mathcal {I}_l\mathcal {I}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))\big |_{l=m=1}= & \left[ \left( \frac{4}{3}rt+\frac{2}{3}r+2t\right) {l^3}{m^3}+\left( 4rt-\frac{2}{3}r+\frac{2}{3}t\right) {l^3}{m^2}\right. \\ & \left. +\left( \frac{3}{2}rt-\frac{1}{4}r+\frac{1}{4}t\right) {l^2}{m^2}\right] \bigg |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}\,^mM_2({\mathbb{G}\mathbb{Y}}_{\beta })= & \frac{41}{6}rt-\frac{1}{4}r+\frac{35}{12}t \end{aligned}$$

Reduced Reverse Redefined Third Zagreb Index of${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$

In view of Table 1, Reduced Reverse Redefined Third Zagreb Index for $\mathcal {RR_D\,M}$-Polynomial is computed as follow

$$\begin{aligned} (\mathcal {D}_l+\mathcal {D}_m)(\mathcal {M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (72rt+36r+108t){l^3}{m^3}+(120rt-20r+20t){l^3}{m^2}\\ & +(24rt-4r+4t){l^2}{m^2} \\ \mathcal {D}_m(\mathcal {D}_l+\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m)= & (216rt+108r+324t){l^3}{m^3}+(240rt-40r+40t){l^3}{m^2}\\ & +(48rt-8r+8t){l^2}{m^2} \\ (\mathcal {D}_l\mathcal {D}_m)(\mathcal {D}_l+\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (648rt+324r+972t){l^3}{m^3}+(720rt-120r+120t){l^3}{m^2}\\ & +(96rt-16r+16t){l^2}{m^2}\\ (\mathcal {D}_l\mathcal {D}_m)(\mathcal {D}_l+\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))\big |_{l=m=1}= & (648rt+324r+972t){l^3}{m^3}+(720rt-120r+120t){l^3}{m^2}\\ & +(96rt-16r+16t){l^2}{m^2}\big |_{l=m=1} \end{aligned}$$

After putting limits, we have

$$\begin{aligned} \mathcal {RR_D}\, ReZG_3({\mathbb{G}\mathbb{Y}}_{\beta };r,t)= & 1464rt+188r+1108t\\ \end{aligned}$$

Reduced Reverse Symmetric Division Degree Index of${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$

In view Table 1, using Eq. (23), we have

$$\begin{aligned} (\mathcal {I}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (4rt+2r+6t){l^3}{m^3}+(12rt-2r+2t){l^3}{m^2}+\left( 3rt-\frac{1}{2}r+\frac{1}{2}t\right) {l^2}{m^2} \end{aligned}$$

After applying operator $\mathcal {D}_l$ (3), we have

$$\begin{aligned} (\mathcal {D}_l\mathcal {I}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (12rt+6r+18t){l^3}{m^3}+(36rt-6r+6t){l^3}{m^2}+(6rt-r+t){l^2}{m^2} \end{aligned}$$

Also from Eq. (21), we get

$$\begin{aligned} \mathcal {D}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (36rt+18r+54t){l^3}{m^3}+(48rt-8r+8t){l^3}{m^2}+(12rt-2r+2t){l^2}{m^2}\\ \end{aligned}$$

After operating integral operator $\mathcal {I}_l$ (5), we have

$$\begin{aligned} \mathcal {I}_l\mathcal {D}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (12rt+6r+18t){l^3}{m^3}+ \left( 16rt-\frac{8}{3}r+\frac{8}{3}t\right) {l^3}{m^2}\\ & +(6rt-r+t){l^2}{m^2}\\ (\mathcal {D}_l\mathcal {I}_m+\mathcal {I}_l\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (24rt+12r+36t){l^3}{m^3}+ \left( 52rt-\frac{26}{3}r+\frac{26}{3}t\right) {l^3}{m^2}\\ & +(12rt-2r+2t){l^2}{m^2} \\ (\mathcal {D}_l\mathcal {I}_m+\mathcal {I}_l\mathcal {D}_m)(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))\big |_{l=m=1}= & (24rt+12r+36t){l^3}{m^3}+ \left( 52rt-\frac{26}{3}r+\frac{26}{3}t\right) {l^3}{m^2}\\ & +(12rt-2r+2t){l^2}{m^2}\big |_{l=m=1}\\ \mathcal {RR_D}\,SDD({\mathbb{G}\mathbb{Y}}_{\beta })= & 88rt-\frac{4}{3}r+\frac{140}{3} \end{aligned}$$

Reduced Reverse Harmonic Index of${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$

In view of Table 1, applying operator (8) on Eq. (19), we have

$$\begin{aligned} {J}\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m)= & (12rt+6r+18t){l^6}+(24rt-4r+4t){l^5}+(6rt-r+t){l^4} \end{aligned}$$

After operating integral operator $\mathcal {I}_l$ (5), we have

$$\begin{aligned} \mathcal {I}_l{J}\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m)= & (2rt+r+3t){l^6}+ \left( \frac{24}{5}rt-\frac{4}{5}r+\frac{4}{5}t\right) {l^5}+\left( \frac{3}{2}rt-\frac{1}{4}r+\frac{1}{4}t\right) {l^4}\\ {2\mathcal {I}_lJ}\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m)= & (4rt+2r+6t){l^6}+ \left( \frac{48}{5}rt-\frac{8}{5}r+\frac{8}{5}t\right) {l^5}+\left( 3rt-\frac{1}{2}r+\frac{1}{2}t\right) {l^4}\\ {2\mathcal {I}_lJ}\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m)\big |_{l=m=1}= & \left[ (4rt+2r+6t){l^6}+\left( \frac{48}{5}rt-\frac{8}{5}r+\frac{8}{5}t\right) {l^5}+\left( 3rt-\frac{1}{2}r+\frac{1}{2}t\right) {l^4}\right] \Bigg |_{l=1}\\ \mathcal {RR_D}\,H({\mathbb{G}\mathbb{Y}}_{\beta })= & \frac{83}{5}rt-\frac{1}{10}r+\frac{81}{10}t \end{aligned}$$

Reduced Reverse Inverse Sum Index of${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$

From Table 1, Harmonic Index for $\mathcal {RR_D\,M}$-Polynomial is

$$\begin{aligned} {\mathcal {D}_l}{\mathcal {D}_m}(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (108rt+54r+162t){l^3}{m^3}+(144rt-24r+24t){l^3}{m^2}+(24rt-4r+4t){l^2}{m^2}\\ {J}{\mathcal {D}_l}{\mathcal {D}_m}(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (108rt+54r+162t){l^6}+(144rt-24r+24t){l^5}+(24rt-4r+4t){l^4}\\ \end{aligned}$$

After operating integral operator $\mathcal {I}_l$ (5), we have

$$\begin{aligned} {\mathcal {I}_l}{J}{\mathcal {D}_l}{\mathcal {D}_m}(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (18rt+9r+27t){l^6}+ \left( \frac{144}{5}rt-\frac{24}{5}r+\frac{24}{5}t\right) {l^5}+(6rt-r+t){l^4}\\ {\mathcal {I}_l}{J}{\mathcal {D}_l}{\mathcal {D}_m}(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))\big |_{l=1}= & (18rt+9r+27t){l^6}+ \left( \frac{144}{5}rt-\frac{24}{5}r+\frac{24}{5}t\right) {l^5}+(6rt-r+t){l^4}\big |_{l=1}\\ \mathcal {RR_D}\,I({\mathbb{G}\mathbb{Y}}_{\beta })= & \frac{264}{5}rt+\frac{16}{5}r+\frac{164}{5}t \end{aligned}$$

Reduced Reverse Augmented Index of${\mathbb{G}\mathbb{Y}}_{\beta }(r,t)$

In view Table 1, applying operator $\mathcal {D}^3_m$ on Eq. (19), we have

$$\begin{aligned} \mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (324rt+162r+486t){l^3}{m^3}+(192rt-32r+32t){l^3}{m^2}+(48rt-8r+8t){l^2}{m^2}\\ \end{aligned}$$

After operating differential operator $\mathcal {D}^3_l$, we have

$$\begin{aligned} \mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (8748rt+4374r+13122t){l^3}{m^3}+(5184rt-864r+864t){l^3}{m^2}\\ & +(384rt-64r+64t){l^2}{m^2} \\ {J}\mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (8748rt+4374r+13122t){l^6}+(5184rt-864r+864t){l^5}\\ & +(384rt-64r+64t){l^4}\\ {Q_{-2}}{J}\mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & {l^{-2}}[(8748rt+4374r+13122t){l^6}+(5184rt-864r+864t){l^5}\\ & +(384rt-64r+64t){l^4}]\\ {Q_{-2}}{J}\mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (8748rt+4374r+13122t){l^4}+(5184rt-864r+864t){l^3}\\ & +(384rt-64r+64t){l^2}\\ \end{aligned}$$

After operating integral operator $I^3_l$, we have

$$\begin{aligned} {\mathcal {I}^3_l}{Q_{-2}}{J}\mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & \left( \frac{2187}{16}rt+\frac{2187}{32}r+\frac{6561}{32}t\right) {l^4}+(192rt-32r+32t){l^3}\\ & +(48rt-8r+8t){l^2}\\ {\mathcal {I}^3_l}{Q_{-2}}{J}\mathcal {D}^3_l\mathcal {D}^3_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))\big |_{l=1}= & \bigg [\frac{2187}{16}rt+\frac{2187}{32}r+\frac{6561}{32}t\bigg ){l^4}+(192rt-32r+32t){l^3}\\ & +(48rt-8r+8t){l^2}\bigg ]\bigg |_{l=1}\\ \mathcal {RR_D}\,A({\mathbb{G}\mathbb{Y}}_{\beta })= & \frac{6027}{16}rt+\frac{907}{32}r+\frac{7841}{32}t \end{aligned}$$

Prediction of the features of graphene nanoribbons

This section describes one of the main focuses of this work, emphasizing the role of topological descriptors in QSPR study and giving instances of its elements of assessment and prediction for boron sheets. An equation derived via regression analysis demonstrated a relationship between topological descriptors and important characteristics of boron sheets. Topological descriptors in QSPR use mathematical models to characterize a Molecular Graph’s properties or functions. Molecular descriptors and the physical properties of chemical compounds are linked by QSPR.

To classify and assess a substance’s properties, a clear and concise strategy must be developed⁶⁹. Consequently, it is critical to investigate and understand the structural properties of molecular compounds. The properties of molecular compounds are analyzed or predicted using modeling techniques or methodologies such as exponential, logarithmic, cubic, linear, and quadratic regression analysis^70,71,72,73. We provide a quadratic regression analysis of the graphene. Apart from its structural and electrical stability, graphene nanoribbons exhibit interesting features related to chemical bonding. Regression analysis was used to examine two-dimensional graphene nanoribbons, such as the $\alpha$-graphyne nanoribbon and the $\alpha$-graphdiyne nanoribbon⁷⁴ and $\beta$-graphyne⁷⁵. Figures 1, 2 and 3 shows an illustration of the graphyne and graphdiyne nanoribbons discussed above. Table 6 provides the reduced reverse degree-vertex value of the graphene nanoribbon’s structure.

In the present research, we examine the graphyne and graphdiyne nanoribbon’s characteristics such as Poisson’s ratio $\mathcal {P^{R}}$ and Young’s modulus $\mathcal {Y^{M}}$, which can be determined using the elastic constant. Table 5 summarizes the results for Young’s modulus $\mathcal {Y^{M}}$, and Poisson’s ratio $\mathcal {P^{R}}$ of different graphyne and graphdiyne nanoribbon’s. The Young’s Modulus show a material’s resistance to changes in length when subjected to tension or compression and Poisson’s ratio measures the deformation (expansion or contraction) of a material perpendicular to the direction of loading.

Table 5 Experimental values of Young’s modulus and Poisson’s ratio of graphene nanoribbons.

Table 6 Computaional values of $\mathcal {RR_D}$ descriptors of graphene and graphdiyne.

Properties analysis through QSPR modeling

Regression modeling is used to examine the mechanical characteristics, Young’s modulus, and Poisson’s ratio of the mentioned graphene derivative’s using topological descriptors. Legendre et al.⁷⁶ and Gauss et al.⁷⁷ established the least squares approach to linear and quadratic regression in 1805 and 1809, respectively. Regression analysis is a statistical technique that determines the correlation between many variables. The correlation coefficient ranges from − 1 to 1. Perfect positive and negative correlation are 1 and − 1, respectively, while near-zero correlation implies inadequate correlation. The equation relating the characteristics and descriptors is derived using regression analysis and a correlation coefficient.

$$\begin{aligned} \mathcal {PCP}= \textbf{p}(\mathcal{T}\mathcal{D})^{2}+\textbf{n}(\mathcal{T}\mathcal{D})+\textbf{m} \end{aligned}$$

where $\mathcal {PCP}$ is the physiochemical property of graphene, $\mathcal{T}\mathcal{D}$ is the topological descriptor, $\textbf{m}$ is the invariant, $\textbf{p}$ and $\textbf{n}$ is the regression coefficient. The correlation coefficient of different topological descriptors has been studied for both chemical attributes. We have concluded that Reduced Reverse degree $\mathcal {RR_D}$ Hyper 2nd Zagrab Index and Reduced Reverse degree $\mathcal {RR_D}$ Redefined Third Zagreb Index has a strong correlation for Young’s modulus and Poison’s ratio respectively. The quadratic regression equations for Young’s modulus are shown as follows:

$$\begin{aligned} \mathcal {Y^{M}}= & 2\times 10^{-6}(\mathcal {RR_D}\,HM_2)^{2}-0.0473(\mathcal {RR_D}\,HM_2)+359.57 \\ \mathcal {Y^{M}}= & 2\times 10^{-6}(\mathcal {RR_D}\,ReZG_3)^{2}-0.052(\mathcal {RR_D}\,ReZG_3)+291.09 \end{aligned}$$

where $\mathcal {Y^{M}}$ is the Young’s modulus and $\mathcal {RR_D}\,HM_2$ is the Reduced Reverse Degree Hyper 2nd Zagrab Index. Similarly, the quadratic regression equations for Poisson’s Ratio are determined as follows:

$$\begin{aligned} \mathcal {P^{R}}= & -7\times 10^{-9}(\mathcal {RR_D}\,HM_2)^{2}+0.0002(\mathcal {RR_D}\,HM_2)-0.6243 \\ \mathcal {P^{R}}= & -1\times 10^{-8}(\mathcal {RR_D}\,ReZG_3)^{2}+0.0002(\mathcal {RR_D}\,ReZG_3)-0.3405 \end{aligned}$$

where $\mathcal {P^{R}}$ is the Poison’s Ratio and $\mathcal {RR_D}\,ReZG_3$ is the Reduced Reverse degree Refefined Third Zagreb Index. Suitable regression models can be used to forecast the molecular features that have a larger dimension. In Fig. 4, the scatter plots corresponding to the most highly associated properties and descriptors are displayed.

Figure 4

Scattering-based visualization of properties and descriptors.

Conclusion

The concept of a $\mathcal {RR_D\,M}$-polynomial based on reduced reverse degrees of a graph is introduced in this paper, and from this, differential and integral operators of a graph are extracted which are helpful in the formulation of reverse degree-based $\mathbb{T}\mathbb{D}$s. We have formulated twelve reverse degree-based $\mathbb{T}\mathbb{D}$s as represented in Table 6 through this methodology. For the structure of the $\alpha$-graphyne, $\beta$-graphyne and $\alpha$-graphdiyne, this study presented the following $\mathcal {RR_D\,M}$-Polynomial.

$$\begin{aligned} \mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m)= & (6rt+3r+9t){l^3}{m^3}+(12rt-6r-6t){l^3}{m^2}\\ \mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m)= & (18rt+r+11t){l^3}{m^3}+(12rt-6r-6t){l^3}{m^2}\\ \mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m)= & (12rt+6r+18t){l^3}{m^3}+(24rt-4r+4t){l^3}{m^2}+(6rt-r+t){l^2}{m^2} \end{aligned}$$

Differential Operators

$$\begin{aligned} \mathcal {D}_l{(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))}= & (18rt+9r+27t){l^3}{m^3}+(36rt-18r-18t){l^3}{m^2} \\ \mathcal {D}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (18rt+9r+27t){l^3}{m^3}+(24rt-12r-12t){l^3}{m^2}\\ \mathcal {D}_l{(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))}= & (54rt+3r+33t){l^3}{m^3}+(36rt-18r-18t){l^3}{m^2}\\ \mathcal {D}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & (54rt+3r+33t){l^3}{m^3}+(24rt-12r-12t){l^3}{m^2}\\ \mathcal {D}_l{(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))}= & (36rt+18r+54t){l^3}{m^3}+(72rt-12r+12t){l^3}{m^2}+(12rt-2r+2t){l^2}{m^2}\\ \mathcal {D}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (36rt+18r+54t){l^3}{m^3}+(48rt-8r+8t){l^3}{m^2}+(12rt-2r+2t){l^2}{m^2} \end{aligned}$$

Integral Operators

$$\begin{aligned} \mathcal {I}_l(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (2rt+r+3t){l^3}{m^3}+(4rt-2r-2t){l^3}{m^2}\\ \mathcal {I}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\alpha };l,m))= & (2rt+r+3t){l^3}{m^3}+(6rt-3r-3t){l^3}{m^2} \\ \mathcal {I}_l(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & \left( 6rt+\frac{1}{3}r+\frac{11}{3}t\right) {l^3}{m^3}+(4rt-3r-3t){l^3}{m^2}\\ \mathcal {I}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{D}}_{\alpha };l,m))= & \left( 6rt+\frac{1}{3}r+\frac{11}{3}t\right) {l^3}{m^3}+(6rt-3r-3t){l^3}{m^2} \\ \mathcal {I}_l(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (4rt+2r+6t){l^3}{m^3}+\left( 8rt-\frac{4}{3}r+\frac{4}{3}t\right) {l^3}{m^2}+\left( 3rt-\frac{1}{2}r+\frac{1}{2}t\right) {l^2}{m^2}\\ \mathcal {I}_m(\mathcal {RR_D\,M}({\mathbb{G}\mathbb{Y}}_{\beta };l,m))= & (4rt+2r+6t){l^3}{m^3}+(12rt-2r+2t){l^3}{m^2}+\left( 3rt-\frac{1}{2}r+\frac{1}{2}t\right) {l^2}{m^2} \end{aligned}$$

We also estimated the physicochemical properties of the $\alpha$-graphyne, $\beta$-graphyne and $\alpha$-graphdiyne by employing the computed reduce reverse degree-based topological descriptors and find the best estimations for the Poisson’s Ratio and Young’s Modulus of the graphene and its derivatives through $\mathcal {RR_{D}}\,HM_{2}$ and $\mathcal {RR_{D}}\,ReZG_{3}$.