Mathematical modeling and topological graph description of dominating David derived networks based on edge partitions

Chemical graph theory is a well-established discipline within chemistry that employs discrete mathematics to represent the physical and biological characteristics of chemical substances. In the realm of chemical compounds, graph theory-based topological indices are commonly employed to depict their geometric structure. The main aim of this paper is to investigate the degree-based topological indices of dominating David derived networks (DDDN) and assess their effectiveness. DDDNs are widely used in analyzing the structural and functional characteristics of complex networks in various fields such as biology, social sciences, and computer science. We considered the FN*, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M}_{2}^{*}$$\end{document}M2∗, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${HM}_{N}$$\end{document}HMN topological indices for DDDNs. Our computations' findings provide a clear understanding of the topology of networks that have received limited study. These computed indices exhibit a high level of accuracy when applied to the investigation of QSPRs and QSARs, as they demonstrate the strongest correlation with the acentric factor and entropy.

www.nature.com/scientificreports/For the sake of simplicity, assume that a and b are two adjacent vertices and E is an edge between them, then the edge partition of E is denoted by E a,b and formulated as E a,b = {η G (a) , η G (b)}.
The degree-based topological indices shows a significant role in the field of mathematical chemistry [3][4][5][6][7] , and widely used to develop models that accurately predict the boiling points of alkanes with carbon atom 8 .Some current discovered degree-based neighborhood indices are presented in 9,10 and shown strong connections between entropy and the acentric factor.
In [11][12][13][14][15][16] , different chemical significant graphs' topological indices are considered.Baig et al. 17 considered the topological indices for several silicates and oxide networks.Ullah et al. 18 , compared and examined the computational characteristics of two carbon nanosheets using some innovative topological indices.The topological characteristics of rhombus-type silicate and oxide networks were explored by Javaid et al. 19 .Recently, Koam et al. 20 , established the entropy measures of Y-junction based nanostructures.Ali et al. 21give some properties of ve-degree based topological indices for hex-derived networks.In this study, an examination was conducted on distance-based topological polynomials that are associated with zero-divisor graphs, as discussed in 22 .The authors of 23 obtained the polynomials of degree-based indices of metal-organic networks.Zaman et al., determined the kemeny's constant and spanning trees of hexagonal ring network 24 .Some upper bound and lower bound of graphs and also the spectral analysis of graphs are discussed in [25][26][27][28] .In this research, inspired by earlier studies, we establish some exact expressions of the different types of Dominating David derived networks and their comparisons.
We have calculated the forgotten index ( F * N ) 29 , the second zargeb index ( M * 2 ) 30 and the Harmonic index ( HM N ) 31 for DDD networks.These topological indices are defined as

Constructions of dominating David derived networks (DDDN)
In the field of chemistry, honeycomb networks are utilized as representations for benzoid hydrocarbons.Honeycomb networks find extensive applications in various domains, including graphics, such as cell phone base stations and image processing.The honeycomb network is formed by enclosing the boundaries with a layer of hexagons.Based on the honeycomb network, different types of Dominating David derived networks can be derived.One can follow the below steps to construct the DDDN (t dimension): Step 1: Consider a t-dimension honeycomb network (see Fig. 1a).
Step 2: Add another vertex to divide each edge into two pieces (see Fig. 1b).www.nature.com/scientificreports/ Step 3: In each hexagonal cell, connect the new vertices by an edge if they are at a distance of 4 within a hexagon (see Fig. 1c).
Step 5: Remove the starting vertices and edges of the honeycomb (see Fig. 1e).
Step 6: Divide each horizontal edge into two parts by addind a new vertex (see Fig. 1f).

Main results
Our key findings rely on the edge partitions of Figs. 2, 3 and 4 as given below.We have calculated these edge partitions based on the degrees of the end vertices of each edge.For instance, the first row of Fig. 1 shows the degrees of the end vertices of edges, while the second row illustrates the count of edges with those specific degrees.
In the same way, we have obtained the other tables.The HM N topological index for DDDN.Let G be a graph in D 1 (t), D 2 (t) and D 3 (t) then according to the definition of HM N and Table 1, we have

The F N
Similarly, from Table 2, we have And from Table 3, we have

Concluding Remarks
In this study, we have considered the F * N , M * 2 and HM N topological indices.Our simulated results help for the better comprehend topology and enhance physical properties of the honeycomb structure.The computed indices, and above, as previously mentioned, have the most closely relates to the acentric factor and entropy consequently, they are extremely accurate in QSPR and QSAR analysis.
In Table 4, the topological indices computed are represented mathematically.As we can see, increasing the values of t, increases the value of the indices as well.We have precise analytical formulations for the D 1 , D 2 and D 3 networks, considering various topological indices.In the rapidly expanding fields of nanotechnology and applications, such as networks, our current discoveries and techniques can be applied to other, more complex structures.The utilization of distance-based topological indices poses greater challenges and complexity, but they can be employed alongside existing methods.Exploring these types of studies will be the focus of future research endeavors.In Table 4 and Fig. 5, we computed the numerical comparison of the certain topological indices for D 1 , D 2 and D 3 networks, which shows that when we increase t as a result the values of the topological indices also increases.These numerical comaprisons also shows that the inceasing rate of HM N for D 3 is greater than the other topological indices.Since, in graph theory, the HM N is a mathematical concept used to describe the connectivity.Therefore, a higher HM N reflects the more connectivity among the atoms of a molecule.This indicates that the D 3 molecule has a greater potential for forming diverse interactions with other molecules and participating in a wider range of chemical reactions.