On the M-polynomial and topological descriptors of the special graph

Introduction

Chemical compounds can be represented effectively through graph-theoretical models, in which the structural characteristics of molecules are translated into mathematical objects. Graph theory provides a powerful framework for investigating the relationship between the structure, properties, and biological activities of chemical compounds. In chemical graph theory, atoms are represented by vertices, while chemical bonds are represented by edges. This graphical representation enables researchers to analyze molecular structures quantitatively and establish relationships between molecular descriptors and various physicochemical properties. Topological indices are numerical descriptors derived from the graph representation of a molecule. These graph invariants play an important role in studying the structural properties of chemical compounds and have found extensive applications in quantitative structure-property relationships (QSPR) and quantitative structure-activity relationships (QSAR). By converting molecular structures into numerical values, topological indices provide an efficient method for predicting chemical and biological properties without performing extensive experimental investigations. In this paper, all graphs under consideration are assumed to be simple, finite, and connected. Let 𝐺 = (𝑉(𝐺), 𝐸(𝐺)) be a graph, where 𝑉(𝐺) denotes the set of vertices and 𝐸(𝐺) denotes the set of edges. If 𝑣 ∈ (𝑉(𝐺)), then the degree of the vertex (𝑣), denoted by 𝑑(𝑣), is defined as the number of edges incident with 𝑣. Over the years, numerous degree-based topological indices have been developed to investigate the structural characteristics of chemical graphs. Among the most widely studied descriptors are the first Zagreb index, the second Zagreb index, the modified Zagreb index, the Randic index, and the harmonic index. These indices have been successfully applied in chemistry, pharmacology, and molecular biology to characterize chemical compounds and predict their physical, chemical, and biological properties [1] .

The concept of the M-polynomial was introduced by Deutsch and Klavžar in 2015 as an efficient mathematical tool for computing degree-based topological indices of chemical graphs [2] . The M-polynomial provides a unified approach for deriving several topological descriptors and has attracted considerable attention in the field of chemical graph theory. Since its introduction, many researchers have investigated its mathematical properties and demonstrated its effectiveness in evaluating various graph invariants associated with molecular structures [3,4] . Several studies have extended the application of the M-polynomial to different classes of chemical graphs. These investigations have shown that the polynomial offers a systematic and convenient method for calculating a wide range of degree-based topological indices, thereby reducing computational complexity and providing deeper insight into the structural characteristics of chemical compounds. In 2017, P. Gayathri and U. Priyanka determined several degree-based topological indices for the zig-zag chain of 8-cycle molecular graphs. Their study included the computation of the Randic index, geometric-arithmetic index, sum-connectivity index, harmonic index, first Zagreb index, second Zagreb index, modified second Zagreb index, inverse sum index, Albertson index, atom-bond connectivity index, symmetric division index, and augmented Zagreb index. The obtained results were expressed in terms of the parameters (a) and (n), where (a) denotes the number of hexagons and (n) represents the number of segments in the molecular graph [5] . In a graph 𝐺, M-polynomial 𝑀(𝐺, 𝑥, 𝑦), is given by

where 𝑚 𝑖𝑗 is total number of edges in 𝐺 such that its end vertices have degree 𝑖 and 𝑗, 𝛿 = 𝑚𝑖𝑛{𝑑(𝑢): 𝑢 ∈ 𝑉(𝐺)} and ∆= 𝑚𝑎𝑥{𝑑(𝑢): 𝑢 ∈ 𝑉(𝐺)}. Is the minimum degree among all the vertices of graph G and is the maximum degree among all the vertices of graph G.

In this paper we have used M-polynomial to compute First Zagreb index 𝑀 1 (𝐺), Second Zagreb index 𝑀 2 (𝐺), the modified Zagreb index 𝑚𝑀 2 (𝐺), Randic ́ index 𝑅 𝛼 (𝐺) and 𝑆𝑆𝐷(𝐺) of a special graph which are defined as follows:

The inverse Randic index 𝑅𝑅 𝛼 (𝐺), Symmetric sum division index 𝑆𝑆𝐷(𝐺) is studied in [3] and defined as follows:

The topological indices can be derived from M-polynomial, which are given in the preceding table: The special graph 𝑆𝐺 𝑛 2 is a planar graph obtained by a recursive path-construction procedure. The construction begins with a path of order 2, which serves as the fundamental building block of the graph. New paths are successively attached to the pendent vertices of the existing graph to generate a sequence of special graphs. In this construction, the graph 𝑆𝐺 0 2 consists of the initial path and is referred to as the root path or level 0 of the construction. A new path is attached to each pendent vertex of 𝑆𝐺 0 2 , producing the graph 𝑆𝐺 1 2 . The path-construction process is then repeated by adjoining new paths to all pendent vertices created at the previous stage. Similarly, the same procedure is applied to 𝑆𝐺 1 2 to obtain 𝑆𝐺 2 2 . At each successive stage, all newly created vertices are joined to form a path on the outer face of the graph, resulting in the next member of the sequence. Repeating this process 𝑛 times generates the family of special graphs 𝑆𝐺 𝑛 2 . Thus, 𝑆𝐺 𝑛 2 is constructed recursively by attaching new paths to the pendent vertices at every level. If the initial path is considered to be at level 0, then the graph obtained after the first iteration corresponds to level 1, the graph obtained after the second iteration corresponds to level 2, and so on. Consequently, after adding new edges or paths to the vertices of the preceding level, a sequence of special graphs 𝑆𝐺 𝑛 2 is obtained.

The recursive nature of this construction makes 𝑆𝐺 𝑛 2 suitable for studying graph invariants and deriving degree-based topological indices through the M-polynomial approach. -To enumerate the number of vertices we develop a counting method. We observe that 2 𝑛+1 levels in graph 𝑆𝐺 𝑛 2 and every 𝑛 𝑡ℎ level has exactly 𝑛 vertices.

Results on Topological Indices of the Special Graph

The total number of vertices on

Also, to enumerate the number of edges we develop approach considering edges at every level and the edges between two preceding levels.

We observe that 2. Next, we define the value of incidence 𝑚 (𝑖,𝑗) in the graph 𝐺 as total number of edges in edge set 𝐸 with end vertex having degree-i and degree-j.

Lemma:

-Let 𝑆𝐺 𝑛 2 be the special graph then the values of incidence of special graph are 𝑚 (1,3) = 2 𝑛+1 where, 𝑛 ≥ 1

And 𝑚 (3,3) = 2 𝑛+1 -3 where, 𝑛 ≥ 1

Proof: -From the graph we observe that at the level 1 and level 2 there are 4 edges such that their end vertices have degree-1 and degree-3.

Also, between the level 𝑛 and level n+1 there are 2 𝑛+1 edges such that end vertices have degree-1 and degree-3.

∴ 𝑚 (1,3) = 2 𝑛+1 where, 𝑛 ≥ 1

Next, we count the number of edges having value of incidence 𝑚 (3,3) at level n+1 there are 2 𝑛+1 -3 edges such that end vertices have degree-3 and degree-3.

∴ 𝑚 (3,3) = 2 𝑛+1 -3 where, 𝑛 ≥ 1 Now for convenience, we denote the M-polynomial of 𝑆𝐺 𝑛 2 by 𝑓(x, y ) i.e., 𝑓(x, y ) = 2 𝑛+1 𝑥𝑦 3 + (2 𝑛+1 -3)𝑥 3 𝑦 3

Next, we compute the operator needed for the estimation of the Next, we compute the operator needed for estimation of DBTI's

(2 𝑛+1 -3)𝑥 3 𝐷 𝑦 𝐷 𝑥 𝑓(x, y ) = [3.

The modified Zagreb index is,

Conclusion

Although explicit formulas provide a direct approach for evaluating topological indices, the present study employs an alternative method based on the M-polynomial to derive various degree-based topological indices of the special graph 𝑆𝐺 𝑛 2 . The Mpolynomial offers a unified and systematic framework for computing these graph invariants and simplifies the evaluation of several important molecular descriptors. The path-construction procedure adopted in this work leads to the generation of a new family of special graphs. This construction can be extended further by repeatedly adjoining new paths to the pendent vertices and forming successive levels of the graph. Consequently, the proposed approach provides opportunities for future investigations involving different path-construction techniques and the development of other classes of recursively defined special graphs. The methodology presented in this paper can also be applied to determine additional topological indices and to explore their potential applications in chemical graph theory and related areas of mathematical chemistry.

Acknowledgement

AcknowledgementsThe authors also extend their heartfelt thanks to Prof. Sandeep G. Kage for introducing them to Graph Theory and guiding them in mathematical research. His valuable teaching, academic support, and research-oriented approach provided a strong foundation for the development of this work. The authors sincerely acknowledge his encouragement and mentorship throughout the preparation of this research article.

References

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