ABC 4 and GA 5 indices of para-line graph of some convex polytopes

In this paper, we will compute Fourth atom-bond connectivity index ABC4(G) and Fifth geometric-arithmetic connectivity index GA5(G), by considering G as para-line graph of some convex polytopes.


Introduction
In the field of chemistry, graph theory has been applied to a wide range of research areas: synthetic chemistry, quantum chemistry, thermochemistry etc. Graph theory has provided the chemist with a variety of very useful tools.Some of the graph theory concepts corresponds to the terms in chemistry e.g. point as an atom, line as a covalent bond, degree as atom valency and path as chemical substructure etc. Topological representation of an object tells us about the number of elements composing it and their connectivity (see [3]).Topological indices are invariant under graph isomorphisms.They have significant role in the quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) investigations (see [4,6,15,35,39]).
Let G be a connected graph with vertex set V (G) and edge set the order of G and q = |E(G)|, the size of G.The degree d v of any vertex v is defined as the number of vertices joining to that vertex v and the degree d e of an edge e ∈ E(G) is defined as the number of its adjacent vertices in V (L(G)), where L(G) is the line graph whose vertices are the edges of G and they are adjacent if and only if they have a common end point in G.In structural chemistry, line graph of a graph G is very useful.The first topological index on the basis of line graph was introduced by Bertz in 1981 (see [5]).For more details on line graph see the articles [12,14,16,17,18,21].The subdivision S(G) of a graph G can be obtained by replacing each edge of G by a path of length 2, or we can say by inserting an additional vertex between each pair of vertices of G.The line graph of subdivision is known as para-line graph.For more details on the topological indicies of para-line graphs we refer to the articles [22],-, [40].
Convex polytopes are fundamental geometric objects.The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimization (see [11]).Also people are paying attention in finding metric dimension and labeling of convex polytopes (see [1,2,19,20]).From these motivational work, we take a step in finding the topological indices of para-line graph of some convex polytopes.

Lemma 1
Let G be a graph with u, v ∈ V (G) and e = uv ∈ E(G).Then: Using above lemma, we can find the degree of a vertex of line graph.
Lemma 2 [13] Let G be a graph of order p and size q, then the line graph L(G) of G is a graph of order q and size 1  2 M 1 (G) − q.

Para-line graphs of convex polytopes D n , Q n and R n
In this section we will discuss the combinatorial aspects of subdivision of some convex polytopes and their para-line graphs.

Convex polytope D n
Consider the graph of convex polytope D n as defined in [1].The convex polytope D n for n = 8 is shown in Figure 1-a.

Convex polytope Q n
Consider the graph of convex polytope Q n as defined in [2].The convex polytope Q n for n = 8 is shown in Figure 2-a.

Convex polytope R n
Consider the graph of convex polytope n as defined in [2].The convex polytope R n for n = 8 is shown in Figure 3-a.

Para-line graph of Convex polytope
is shown in Figure 3-c.Using Lemma 2, the total number of edges are 25n.Also |V (L(S(Q n )))| = 12n in which 3n vertices are of degree 3, 4n vertices are of degree 4 and 5n vertices are of degree 5.

The edge partitions of para-line graph of convex polytopes w.r.t degree sum
For a vertex u ∈ V (G), let S u = ∑ uv∈E(G) d v is the degree sum of u.For uv ∈ E(G), S u and S v is the sum of degrees of all neighbors of vertex u and v in G respectively.We partition E(G) into subsets based on the degree sum of the end vertices of edges in G.The edge partition of L(S(D n )), L(S(Q n )) and L(S(R n )) with respect to degree sum are shown in Tables 1, 2 and 3 respectively.

Topological Indices of para-line graphs of some Convex Polytopes
In this section we will compute Fourth Atom-Bond Connectivity Index and Fifth Geometric-Arithmetic Index of the para-line graphs of some convex polytopes discussed in the first section.

Fourth Atom-Bond Connectivity Index
M. Ghorbani et al. in [7,8,9] proposed Fourth Atom-Bond connectivity index as: where S u is the sum of degrees of all neighbors of vertex u in G.In other words, S u = ∑ uv∈E(G) d v .Similarly for S v .

Fifth Geometric-Arithmetic Index
This index was introduced by Graovac et al. in [10] as: Theorem 1 Let L(S(D n )), L(S(Q n )) and L(S(R n )) are the para-line graphs of convex polytopes D n , Q n and R n respectively then: ABC 4 (L(S(D n ))) = 8n.ABC 4 (L(S(Q n ))) =
D n We obtain the graph S(D n ) by replacing each edge of D n by a path of length 2. The subdivision of D n for n = 8 is shown in Figure 1-b.Using Lemma 2, the total number of edges are 12n.Also |V (S(D n ))| = 10n in which 6n vertices have degree 2 and 4n vertices have degree 3.1.2.2.Para-line graph of Convex polytopeD n The para-line graph L(S(D n )) of D n for n = 4 is shown in Figure 1-c.Using Lemma 2, the total number of edges are 18n.Also |V (L(S(D n )))| = 12n and all vertices are of degree 3.
R n We obtain the graph S(R n ) by replacing each edge of R n by a path of length 2. The subdivision of R n for n = 8 is shown in Figure3-b.Using Lemma 2, the total number of edges are 12n.Also |V (S(R n ))| = 9n in which 6n vertices have degree 2, n vertices have degree 3, n vertices have degree 4 and n vertices have degree 5.

Table 1 .
The edge partition of para-line graph of Dn w.r.t degree sum

Table 2 .
The edge partition of para-line graph of Qn w.r.t degree sum(S u , S v ) Number of edges (S u , S v ) Number of edges

Table 3 .
The edge partition of para-line graph of Rn w.r.t degree sum