Derivation of mathematical closed form expressions for certain irregular topological indices of 2D nanotubes

A numeric quantity that characterizes the whole structure of a network is called a topological index. In the studies of QSAR and QSPR, the topological indices are utilized to predict the physical features related to the bioactivities and chemical reactivity in certain networks. Materials for 2D nanotubes have extraordinary chemical, mechanical, and physical capabilities. They are extremely thin nanomaterials with excellent chemical functionality and anisotropy. Since, 2D materials have the largest surface area and are the thinnest of all known materials, they are ideal for all applications that call for intense surface interactions on a small scale. In this paper, we derived closed formulae for some important neighborhood based irregular topological indices of the 2D nanotubes. Based on the obtained numerical values, a comparative analysis of these computed indices is also performed.

www.nature.com/scientificreports/ The Zagreb indices, including the first and second Zagreb indices, measure the sum of the vertex degrees and the product of vertex degrees, respectively [19][20][21] . These degree-based indices have been successfully applied in chemistry, network analysis, and mathematical chemistry. Variants of Zagreb indices, such as the geometricarithmetic indices and the atom-bond connectivity indices, have been developed to enhance their discriminatory power [22][23][24] . Randic-type indices, such as the augmented Zagreb index, the Randic connectivity index, and the atom-bond connectivity indices are derived from degree sequences and capture information regarding vertex degrees 25 . These indices have found applications in chemical graph theory, network analysis, and bioinformatics 26,27 .
Degree-based topological indices have found numerous applications across different disciplines, including chemistry, biology, materials science, and social network analysis. They have been utilized for drug design, chemical property prediction, molecular structure-property relationships, protein classification, community detection, and modeling complex networks [28][29][30] .
Recent research has focused on developing new degree-based topological indices with enhanced discriminative capabilities and exploring their applications in emerging areas, such as social networks, biological networks, and complex systems. Efforts have also been made to combine degree-based indices with other topological indices to capture more comprehensive structural information. Future directions involve investigating the theoretical properties of degree-based indices, developing efficient algorithms for their computation, and exploring their applications in further real-world problems [31][32][33] .
The application of Quantity Structure Activity Relationship (QSAR), which links biological structure and activity with certain constraints and properties of molecules as a result, is extensive in biology as well as in the pharmaceutical and medical fields 34,35 . Carbon nanotubes have an intriguing role because of its special application in chemical sciences. The chemical graph theory has found significant role in thousands of topological indicators. The irregularity topological indices are listed in Table 1.
Motivated by the above formulas, we have introduced some new neighborhood version of irregular topological indices in Table 2.
Numerous efforts have been made to investigate the topological indices for various nanotubes and nanosheets in the literature. The topological invariants of Pent-Heptagonal nanosheets and TURC 4 C 8 (S) are studied respectively in 44,45 . The topological indices of V-phenylenic type nanotori and nanotubes have been discussed in 46 , and armchair polyhex type nanotube in 47 . For detailed insights into the investigations on topological modeling and analysis of micro and nanostructures, one might consult refs 27,30,32,[48][49][50][51][52][53][54][55][56][57][58][59][60][61][62] . Despite all these investigations, the Nano structural topology has not yet been unveiled completely. In this study, we derived closed formulae for some neighborhood version of irregular topological indices of the nanotubes HAC 5 C 7 [p, q] and HAC 5 C 6 C 7 [p, q] , and performed a comparative analysis based on the numerical results.

The HAC 5 C 7 [p, q] nanotubes (p, q > 1)
A trivalent adornment has remained complete by joining C 5 and C 7 and recognized as C 5 C 7 net. It has been utilized to conceal both a tube and a torus. As a C 5 C 7 net, the HAC 5 C 7 [p, q] nanotube can be studied. In 2007, Iranmanesh and Khormali calculated the vertex-Szeged index of HAC 5 C 7 nanotube. The two dimensional lattice of HAC 5 C 7 has been explained consistently. In the entire lattice, the number of heptagons and period are represented by p and q in row. There are 8pq + p vertices and 12pq − p edges, respectively . The three rows of HAC 5 C 7 is said to be m th period (Fig. 1). Consider the graph of HAC 5 C 7 is represented by G . The cardinality of vertex set is 8pq + p and edge set is 12pq − p for the graph G. The vertex set is divided into three categories based on their degrees. The order of vertex V 1 is 8pq. Similarly, |V 2 | = 2p + 2 , |V 3 | = 8pq − p − 2 . In the whole study, we denote Table 1. List of the irregular topological indices.

Introduced by Notation Formula
In 36 , Albertson defined the Albertson index (AL) AL(G) uvǫE |d u − d v | Vukicevic and Gasparov defined the IRL index in 37 IRL(G) uvǫE |lnd u − lnd v | Abdo et al. defined the total irregularity index (IRRT) in 38 IRRT(G) The Randić index (Li and Gutman) 40 www.nature.com/scientificreports/ the adjacent vertices by p and q, i.e. pq ∈ E G . The edge set is divided into the subsequent sections according to their sum of neighborhood degree, called the frequency, which is shown in Table 3.
Proof By definition of N AL (G) and from the neighborhood edge partitions in Table 3, one has Proof Similar to the proof of theorem 1, one has Table 2. List of the neighborhood version of irregular topological indices.

Notation Formula
Proof Based on Table 3 and the definition of N IRRT we have Proof Together Table 3 with the definition N IRF (G) = pqǫE δ p − δ q 2 , one has www.nature.com/scientificreports/

The HAC 5 C 6 C 7 [p, q] nanotubes (p, q > 1)
Let G be the graph of HAC 5 C 6 C 7 p, q nanotube. Then,

Theorem 12
Assume that G ∈ HAC 5 C 6 C 7 p, q be a graph as shown in Fig. 2. Then, N AL (G) = 18p Proof By definition of N AL (G) and Table 4 one has: www.nature.com/scientificreports/ Theorem 13 Assume that G ∈ HAC 5 C 6 C 7 p, q be a graph as shown in Fig. 2. Then, N IRL (G) = 9p Proof By definition N IRL (G) = pq∈E |lnδ p − lnδ q | Theorem 14 Assume that G ∈ HAC 5 C 6 C 7 p, q be a graph as shown in Fig. 2. Then, N IRRT (G) = 9p Proof By definition N IRRT (G) = 1 2 pqǫE δ p − δ q Theorem 15 Assume that G ∈ HAC 5 C 6 C 7 p, q be a graph as shown in Fig. 2.
Theorem 16 Assume that G ∈ HAC 5 C 6 C 7 p, q be a graph as shown in Fig. 2. Then, N IRA (G) = 0.0184628432p  Table 4. The neighborhood edge partitions of HAC 5 C 6 C 7 nanotube.

Theorem 18
Assume that G ∈ HAC 5 C 6 C 7 p, q be a graph as shown in Fig. 2. Then, N IRLF (G) = 2.510484p Proof By definition N IRLF (G) = pqǫE

Theorem 22
Assume that G ∈ HAC 5 C 6 C 7 p, q be a graph as shown in Fig. 2. Numerical discussion and conclusion. In this section, we conclude our work with some important remarks. In Section "The HAC 5 C 7 [p, q] nanotubes (p, q > 1)" we constructed the structures of HAC5C7[p, q] nanotubes for p, q > 1 . Based on Fig. 1a, b, we obtained the neighborhood edge partitions as shown in Table 3.
With the help of these partitions, we determined the neighborhood irregularity topological indices. Moreover,  www.nature.com/scientificreports/ the numerical and graphical comparisons among all considered topological indices are given in Table 5 and Fig. 3. Which shows that there is a positive relation between p, q and these topological indices. That is to say, when we increase the values of p and q the values of topological indices also increase. Hence, from this comparison it is easy to see that the value of N IRF index is higher than the values of remaining topological indices. In Section "The HAC 5 C 6 C 7 [p, q] nanotubes (p, q > 1)", we constructed the structures of HAC 5 C 6 C 7 p, q nanotubes for p,q > 1. Based on Fig. 2a, b, we obtained the edge partitions as shown in Table 4. With the help of these edge partitions, we determined the neighborhood irregularity topological indices. Moreover, the numerical and graphical comparisons among all considered topological indices are given in Table 6 and Fig. 4. Which shows that there is a positive relation between p, q and these topological indices, when we increase the values of p and q, the values of topological indices also increase. Hence, from this comparison it is easy to see that the value of N IRF index is higher than the values of remaining topological indices.

Data availability
All data generated or analysed during this study are included in this article.