Topological invariants for the line graphs of some classes of graphs

Abstract Graph theory plays important roles in the fields of electronic and electrical engineering. For example, it is critical in signal processing, networking, communication theory, and many other important topics. A topological index (TI) is a real number attached to graph networks and correlates the chemical networks with physical and chemical properties, as well as with chemical reactivity. In this paper, our aim is to compute degree-dependent TIs for the line graph of the Wheel and Ladder graphs. To perform these computations, we first computed M-polynomials and then from the M-polynomials we recovered nine degree-dependent TIs for the line graph of the Wheel and Ladder graphs.


Introduction
In mathematical chemistry, we use mathematics to solve problems of chemistry, and a key area of research in mathematical chemistry is Chemical graph theory in which we represent compounds and chemical structures with graphs and apply graph theory to study their topologies. Topological indices (TIs) are real numbers attached to graph networks and graph of compounds.
TIs remain invariant and can be used in predicting the properties of interesting compounds [1].
In the field of heminformatics, quantitative structureactivity relationship (QSAR) and quantitative structureproperty relationship (QSPR), together with Tis, are utilized to study properties and chemical bioactivity of compounds [2]. Like TIs, polynomials also support a considerable number of applications in network theory and chemistry; for instance, the Hosoya polynomial, which is also known as Wiener polynomial [3], is helpful in constructing distance-dependent TIs. The M-polynomial was introduced previously [4] for deciding degree-dependent TIs [5,6]. Definition 1. For a simple connected graph G, the M-polynomial is defined in [4] as: The Wiener polynomial was the first such function, with the Wiener index being introduced in 1947 [7]. Thus, we can say that Harold Wiener began the theory of TIs [8,9]. After Wiener's work, Milan Randic [10] introduced the first degree-dependent TI, which is today known as Randic index (RI), in 1975. The mathematical formula of RI is In 1988, a generalized version of RI was defined by several researchers [11,12]. This version attracted the attention of both mathematicians and chemists [13]. Numerous numerical properties of this simple TI have studied, and results are presented in research reports [14] and a helpful book [15]. In addition, many research papers and books [16][17][18] have been published regarding RI. Two reviews of RI were written by Randic [19,20] and three more reviews have been written on this TI by other scientists [21][22][23].
After RI, the most interesting TIs are 1 st Zagreb index (ZI) and 2 nd ZI [24][25][26][27]. The first and second ZIs were proposed by Gutman and Trinajstic' and are defined as In the remaining paper, we consider G to be the simple connected graph. A graph G with vertex set V(G) and edge set E(G) is connected if there exists a connection between any pair of vertices in G [33]. The quantity of vertices of G adjoining a given vertex v, is the "degree" of this vertex and will be denoted by d v . Throughout this paper, G will represent a connected graph, V its vertex set, E its edge set, and v d the degree of its vertex v. The line graph of G is denoted by L(G) and is obtained from G by associating a vertex with each edge of the graph and connecting two vertices with an edge if and only if the corresponding edges of G have a vertex in common.
In this paper we study line graph of Wheel and Ladder graphs. We computed several degree-based topological indices of the understudy families of graphs.

Methodology
There are three kinds of TIs: 1. Degree-based TIs 2. Distance-based TIs 3. Spectral-based TIs In this paper, we aim to compute degree-dependent TIs. To compute degree-based TIs of line graph of Wheel, Ladder and Bipartite graphs, we first drew line graphs and then we divided the edge sets of these line graphs into classes based on the degree of the end vertices and computed their cardinality. From this edge partition, we computed our desired results. First, we computed M-polynomials of the understudy families of graphs. Then, by applying calculus and using table 1, we computed several TIs.
The relationship between M-polynomial and indices is presented in table 1 [4] where x

Main Results
This section consists of two subsections. In the first subsection we study the line graph of the Wheel graph, and in the second subsection we study the line graph of the ladder graph.

M-polynomial of line graph of Wheel Graph
In order to construct a wheel graph, we connected a single vertex to other vertices in a cycle. A wheel graph with n vertices can also be defined as the 1-skeleton of an (n-1)-gonal pyramid. The Wheel graph is given in Figure 1 and its line graph is given in Figure 2.

Theorem 1
Assume G to be the line graph of Wheel graph; then, we have

Proof
The line graph of Wheel graph is shown in Figure 2. From Figure 2, we have We can divide the vertex set of G into the following two types, depending on the degree We can divide the edge set of G into the following three classes depending on each edge at the degree of end vertices :

M-polynomial of the line graph of the Ladder Graph
A Ladder graph is a planar undirected graph with 2n vertices and 3n-2 edges. The ladder graph can be obtained as the Cartesian product of two path graphs, one of which has only one edge. In this section, let G denote the line Graph of Ladder Graph. The line graph of ladder graph is given in Figure 3.

Theorem 3
Let G be the line graph of Ladder graph. Then we have x y x y x y n M G x y x y x y n x y n

Case 1 when n=2
We can divide the edge set of the line graph of ladder graph into following three classes depending on each edge at the end vertices of the degree

M G x y E G x y E G x y E G x y
x y x y x y = + + + +

Case 2 when n>2
We can divide the edge set of the line graph of ladder graph into following three classes depending on the degree of end vertices of each edge: From the definition of M-polynomial, we have

Proposition 2
Let G be the line graph of Ladder graph.

Conclusions
TIs are numbers associated with the molecular graphs of chemical structures that are useful in predicting properties of chemical compounds of interest [33][34][35][36][37][38][39]. TIs and QSARs together are used in chemistry, and they tell us about the topology of compounds under study. Calculating TIs of molecular graphs of chemical structures is an interesting problem and has attracted many researchers in recent years. In this paper, we computed M-polynomials for the line graph of some interesting families of graphs. We also computed different TIs from the computed M-polynomials by applying fundamental results of Calculus. We computed Zagreb indices, Randic indices, Symmetric division index, inverse sum index, etc. Our results are applicable in predicting properties of compounds. For example, the symmetric division index is a good predictor of the total surface area, Zagreb indices are used to calculate total pi-electronic energy, the inverse sum index is helpful in approximation of total surface area, augmented Zagreb index is a good predictor of the heat of formation, and the harmonic index is used for medication configuration.

Conflict of interest:
Authors declare no conflict of interest.