Topological Indices of Para-line Graphs of V-Phenylenic Nanostructures

Abstract The degree-based topological indices are numerical graph invariants which are used to correlate the physical and chemical properties of a molecule with its structure. Para-line graphs are used to represent the structures of molecules in another way and these representations are important in structural chemistry. In this article, we study certain well-known degree-based topological indices for the para-line graphs of V-Phenylenic 2D lattice, V-Phenylenic nanotube and nanotorus by using the symmetries of their molecular graphs.


Introduction
Chemical graph theory is a eld of mathematical chemistry in which we implement the tools from graph theory to model chemical aspects mathematically. It is recorded in [1,2] that the structure of a molecule is strongly related to its chemical properties such as strain energy, boiling point and heat of formation. Molecular graphs can be used to model the molecules and molecular compounds by considering atoms as vertices and the chemical bonds between the atoms as edges. Topological index (TI) is a kind of numerical graph invariant which is used to correlate the physical and chemical properties of a molecular graph. In this sense, topological indices perform a signi cant role in chemical graph theory. Consider the molecular graph G having vertex set V G and edge set E G . Let I Gp be the set of edges of G that are incident with a vertex p ∈ V G , then the degree, dp , of p is de ned as the cardinality of set I Gp and Sp = q∈Np dq , where dq is the degree of vertex q and the set Np consists of all neighbor vertices of p i.e. Np = {q ∈ V G |pq ∈ E G }. For any natural number t, we de ne V t = {p ∈ V G | Sp = t}. The subdivided graph of G is denoted by S(G) and de ned by replacing each of its edge with the path having length . The line graph of G is symbolized by L(G). This graph is constructed by taking the vertex set V L(G) = E G and the edge set E L(G) which has the property that for two vertices p, q ∈ V L(G) , pq ∈ E L(G) ⇐⇒ p, q ∈ E G have a common vertex. The line graph of subdivided graph L(S(G)) is termed as the para-line graph of G. Para-line graphs are used to understand the structure of a molecular graph and in this sense they receive much attention in structural chemistry. The atomic hybrid orbitals in a molecular graph corresponds to the vertices of its para-line graph and the strong links between the pairs of these orbitals correspond to the edges of its para-line graph. Klein et al. [3] presented some applications and basic properties of the para-line graphs in chemical graph theory. Generally, topological indices can be categorized in three classes: degree-based, distance-based and spectrum-based indices. Among them, degree-based indices have great applications in chemical graph theory [4,5] and they can be de ned in two ways as where the sum runs over all pairs of adjacent vertices of G and F = F(x, y) is a suitably selected function. Milan Randić proposed in 1975 a structural descriptor called the branching index [6] which is applicable for rating the degree of branching of the carbon-atom skeleton of saturated hydrocarbons. This index was renamed as the Randić connectivity index, which is de ned as the sum of the Randić weights (dp dq) − for all edges. The generalization of the Randić index for any real number α, is termed as the general Randić connectivity index and is de ned by taking F = (dp dq) α in equation (1). Li and Zhao presented the rst general Zagreb index in [7], which is de ned as Mα(G) = p∈V(G) (dp) α . The sum-connectivity index was presented in [8] and was modi ed to the general sum-connectivity index in [9], which is formulated by selecting F = (dp + dq) α in equation (1). It is recorded in [10] that the general Randić connectivity and sum-connectivity indices correlate greatly with the π-electron energy of benzenoid hydrocarbons. Estrada et al. [11] presented the atom bond connectivity index (ABC). This index is de ned by choosing F = (dp + dq − )/dp dq in equation (1). D. Vukičević and B. Furtula [12] proposed the geometric-arithmetic (GA) index that is de ned by setting F = dp dq /(dp + dq) in equation (1). The fourth ABC index was presented by Ghorbani and Hosseinzadeh [13] and is de ned by choosing F = (Sp + Sq − )/Sp Sq in equation (2). The fth GA index (GA ) was presented by Graovac et al. [14] which is de ned by setting F = Sp Sq /(Sp + Sq) in equation (2). The explicit expressions of Zagreb indices for the para-line graphs of ladder, tadpole and wheel graphs, was presented by Ranjini et al. [15]. Su and Xu [16] studied general sum-connectivity indices for these para-line graphs. Nadeem et al. [17] presented the ABC and GA indices for these para-line graphs. In [18], they also studied ABC, ABC , GA, GA , general Zagreb, generalized Randić and general sum-connectivity indices for the para-line graphs of D-lattice TUC C (R), TUC C (R) nanotube and TUC C (R) nanotorus. Recently, Akhter et al. [19] and Mufti et al. [20] computed ABC, ABC , GA, GA , rst general Zagreb, general sum-connectivity and general Randić connectivity indices for the para-line graphs of certain benzenoid structures. In this paper, we present these indices for the para-line graphs of V-phenylenic D-lattice, Vphenylenic nanotube and nanotorus.

V-Phenylenic Nanostructures
The Phenylenes belong to the family of polycyclic non-benzenoid alternate conjugated hydrocarbons in which the carbon atoms form hexagons and squares. Each square is adjacent to two detached hexagons. From this, some larger compounds can be formed such as V-phenylenic D lattice, V-Phenylenic nanotube and nanotorus.
Let TUC C C [m, n] represents the V-phenylenic nanostructures where m denotes the number of hexagons in a row and n denotes the number of rows of hexagons in V-Phenylenic D-lattice, V-Phenylenic nanotube and nanotorus as presented respectively in Figure 1 (a), (b) and (c). The order and size of these nanostructures are given in Table . (a) (b) (c)

Main Results
In this section, we derive the topological indices for the para-line graphs of V-Phenylenic nanostructures by using their symmetric structures. The para-line graphs of these structures are presented in Figure 2 We use cardinalities of partite sets given in Table 2 and by choosing the corresponding function F(dp , dq) in   Table 3. Table 3 and choosing the corresponding function F(Sp , Sq) in equation (2), we get the required results.

By using
.

TI's of the para-line graph of TUC C C [m, n] nanotube
Proof. The para-line graph H of TUC C C [m, n] nanotube and is presented in Figure 2 (b). One can easily verify that |V H | = ( pq−p). Among them, there are m and m( n− ) vertices of degree and respectively. By using the handshaking lemma, we have Therefore, we get the following disjoint edge partite subsets of E H and present its cardinalities in Table 4.
We apply equation (1) to the information in Table 4 by choosing the corresponding functions F(dp , dq) and get the desired results.  Table 5. We apply equation (2) to Table 5 by taking the corresponding function F(Sp , Sq) and get the desired indices.
Proof. The para-line graph of TUC C C [m, n] nanotorus and its para-line graph K is presented in Figure  2 (c). One can easily check that in K, |V K | = mn and all these vertices are of degree 3. By using the handshaking lemma, we have |E K | = mn. So, we have exactly one edge partition of E K which is given by With this cardinality, we apply equation (1) by setting the corresponding function F(dp , dq) and get the desired indices.

Conclusion
In this article, well-known degree-based topological indices such as rst general Zagreb, general Randić connectivity, general sum-connectivity, ABC, ABC , GA and GA indices are studied. These indices correlate many chemical properties such as stability, heat of formation, boiling point and strain energy of chemical compounds. By using the symmetric structure property of V-phenylenic nanostructures, we present these indices for their para-line graphs which will help the people to interpret and analyze the underlying topologies of these nanostructures.