Padmakar-Ivan index of some types of perfect graphs

The Padmakar-Ivan (PI) index of a graph G is deﬁned as PI ( G ) = (cid:80) e ∈ E ( G ) ( | V ( G ) | − N G ( e )) , where N G ( e ) is the number of equidistant vertices for the edge e . A graph is perfect if for every induced subgraph H , the equation χ ( H ) = ω ( H ) holds, where χ ( H ) is the chromatic number and ω ( H ) is the size of a maximum clique of H . In this paper, the PI index of some types of perfect graphs is obtained. These types include co-bipartite graphs, line graphs, and prismatic graphs.


Introduction
All graphs considered in this paper are finite, simple and connected. For a graph G, the distance between two vertices x, y is denoted by d (x, y) . A vertex w is equidistant for an edge e = xy if d (x, w) = d (y, w) . For an edge e ∈ E (G), denote by D G (e) the set of all equidistant vertices in G. In particular, D i (e) denotes the set of vertices at distance i for e. Also, we denote |D G (e)| = N G (e).
The vertex Padmakar-Ivan (PI) index of a graph G is a topological index, defined as P I (G) = e=uv∈E(G) (n u (e) + n v (e)) , where n u (e) denotes the number of those vertices of G whose distance from the vertex u is smaller than the distance from the vertex v and n v (e) denotes the number of those vertices of G whose distance from v is smaller than the distance from u. Since n u (e) + n v (e) = |V (G)| − N G (e), the PI index can be rewritten as The PI index was proposed by Khadikar [10] in 2000. Khadikar and his coauthors investigated the chemical and biological applications of this index in [11]. Khalifeh [12] introduced a vertex version of the PI index and using this notion, they computed exact expression for the PI index of Cartesian product of graphs. John and Khadikar established a method for calculating the PI index of benzenoid hydrocarbons using orthogonal cuts in [9]. Gutman and Ashrafi [6] obtained the PI index of phenylenes and their hexagonal squeezes. The PI index of bridge graphs and chain graphs was studied in [13]. Das and Gutman [3] obtained a lower bound on the PI index of a connected graph in terms of the number of vertices, edges, pendent vertices, and the clique number, and also they characterized the extremal graphs. There are different types of topological indices; for example distance-based topological indices, degree-based topological indices, etc. Topological indices has many applications in the field of mathematical chemistry. Trinajstić and Zhou introduced the sum-connectivity index and found several basic properties in [16]. Many topological indices and their applications are thoroughly explored in [15]. Ilić and Milosavljević introduced the weighted vertex PI index and established some of its basic properties in [7]. The weighted PI index of a graph G is given as Gopika et al. [5] obtained the weighted PI index of the direct and strong product for certain types of graphs. Indulal et al. [8] studied the graphs satisfying the equation P I (G) = P I (G − e).
A graph is perfect if for every induced subgraph H, the equation χ (H) = ω (H) holds, where χ (H) is the chromatic number and ω(H) is the size of a maximum clique of H. A claw-free graph is a graph in which no vertex has three pairwise nonadjacent neighbours. Every claw-free graph is a perfect graph. A survey on claw-free graphs is given in [4]. Chudnovsky and Seymour studied the structure of claw-free graphs thoroughly in a series of seven papers from 2007 to 2012. For example, in the first paper [1] of this series, they studied the orientable prismatic graphs and in the second paper [2], they studied non-orientable prismatic graphs. In this paper, we obtain the PI index of some classes of perfect graphs, including co-bipartite graphs, line graphs, and prismatic graphs.

Co-bipartite graphs
An edge e = xy of a graph G is said to be an equidistant edge for a vertex a ∈ V (G) if d (a, x) = d (a, y). The edge e is at distance r for a vertex a if d (a, x) = d (a, y) = r. The set of all equidistant edges of a is D G (a) = {e = xy ∈ E (G) : d (a, x) = d (a, y)} and we take N G (a) = |D G (a)|. It is easy to see that e∈E(G) N G (e) = a∈V (G) N G (a) . Proof.
Let G (U, V ) be a bipartite graph with partite sets U and V . A co-bipartite graph is the complement of a bipartite graph G(U, V ) and it is denoted as G. In G, the vertices in U and the vertices in V forms two disjoint cliques. Every co-bipartite graph is a perfect graph. The diameter of a connected co-bipartite graph is either 2 or 3.
Consider a bipartite graph G (U, V ) with |U | = n and |V | = m. Let ∆ 1 and ∆ 2 be the maximum degree in U and V respectively, where ∆ 1 ≤ m and ∆ 2 ≤ n . Let provided that the degree of every vertex in V 1 is less than n and the degree of every vertex in V 2 is n.
The degrees in G (see Figure 1) are given as We partition E G with E 1 , E 2 and E 3 , where E 1 is the set of edges in the clique with vertices in U , E 2 is the set of edges in the clique with vertices in V and For a vertex u ∈ U , it is easy to see that A vertex v i ∈ V 1 has (n − g i ) neighbours in U and the remaining g i vertices are at distance 2, which means that Similarly, a vertex v ∈ V 2 has no neighbours in U and Also, (2) Now, we combine the three equations (1), (2), and (3) to calculate P I G . 1).
A bipartite graph G (U, V ) is (x, y)-biregular if each vertex in U has degree x and each vertex in V has degree y.

Corollary 2.3. If G is a k-regular bipartite graph with 2n vertices then
Proof. We know that the weighted PI index of a regular graph is a multiple of its PI index. Therefore, P I w G = 2 (2n − k − 1) P I G = 4n (2n − k − 1) 2n (k + 1) − 2k 2 + k + 1 .

Line graphs of some classes of graphs
Let G be a graph with n vertices and m edges. Its line graph denoted by L (G), is a simple graph whose vertices are the edges of G and two vertices are adjacent in L (G) if the corresponding edges are adjacent in G. Let T be a tree with n vertices. Every vertex v in T with degree i, i > 2, forms a star K 1,i in T , we denote it by S i . Let S be the collection of all stars in T . If we delete edges of all stars in T , the remaining edges of T are parts of paths. Some paths have both of its end vertices common with the stars; we call them as central paths and the remaining have one end vertex shared with stars (paths) and the other end vertex is a pendent vertex; we call them leaf paths. We denote the central path with the length l by P l and pendent path with the length l by P * l . As we know that line graphs of stars are complete graphs and line graphs of paths are paths. Each star S i in T is transformed to a clique with K i in L (T ). The central path P l has l edges, so it is transformed to the path with l vertices having length l − 1 and each of its end vertices is connected with a vertex of a clique in L(T ), so it has l − 1 + 2 = l + 1 edges. Each leaf path P * l is transformed to a path with l vertices and l − 1 edges, and it is connected with a vertex of L(T ), so it has l edges. Proof. Let T be a tree with n vertices. Assume that the edge set E(T ) is the union of m stars S ki , r central paths P fi , and s pendant paths P * gi . Let us assume that Then, We claim that Let e be an edge of K ki in L(T ) and let v ∈ V (K ki ) be equidistant to e. If we delete all the edges of K ki , then L(T ) has more than one component. All vertices in the component W containing v are also equidistant to e. If we consider all the edges and vertices of K ki , then Also, since each edge of a path is a cut edge, there is no equidistant vertex corresponding to those edges. Each P fi+1 contributes (f i + 1)(n − 1) and each P * gi contributes g i (n − 1) to the PI index of L(T ). Thus, Since each P fi lies between two S ki , it holds that r = m − 1. Therefore, P I (L(T )) = (n − 1) (n − 1 − 1) = (n − 1) (n − 2) = P I (T ) − 2 (n − 1) .
Let K n be the complete graph with n vertices. The graph L (K n ) is the edge disjoint union of n cliques A 1 , A 2 , A 3 , ..., A n , each of which has order n − 1. Also, each vertex of L (K n ) is a part of exactly two cliques and any two cliques in L (K n ) have exactly one vertex in common.
Proof. The edge set of L (K n ) can be partitioned as where A i s are cliques of order n − 1. Let e = uv be an arbitrary edge in L (K n ), then e ∈ A i for some i. All the remaining vertices in A i are at distance one, so V (A i ) \ {u, v} ⊆ D 1 (e). Since each vertex belongs to exactly two cliques, u ∈ A j and v ∈ A h , for some i ∈ {j, h}. Also, two cliques have exactly one vertex in common, say w, which is different from u and v. So, d(u, w) = d(v, w) = 1 implies that w ∈ D 1 (e). Moreover, the number of vertices at distance 2 is Therefore, Next, we consider the complete bipartite graph K n,m = G(U, V ) with |U | = n and |V | = m. Its line graph L(G) is the edge disjoint union of m + n cliques, where m cliques have order n and n cliques have order m. Each vertex in L(G) belongs to exactly two cliques, one of which has order n and the other is of order m. Two cliques of the same order have no vertex in common.

Prismatic graphs
Chudnovsky and Seymour studied different structural properties of claw-free graphs in a series of seven papers. In their first paper [1] of this series, they studied the orientable prismatic graphs and in the second paper [2] they studied nonorientable prismatic graphs. A graph G is prismatic if for every triangle T in G, every vertex not in T has exactly one neighbour in T . Core of a prismatic graph is the union of all triangles in G. Total coloring of prismatic graphs are discussed in [14]. Here, we consider a particular class of prismatic graphs, namely rigid prismatic graphs. A prismatic graph G with core W is rigid if • there does not exist two distinct vertices u and v, not in the core, with the same neighbouring set in W , • every two non-adjacent vertices have a common neighbour in the core. Proof. Let G be a rigid prismatic graph with p triangles, n vertices, and m edges. Since every two non-adjacent vertices of G have a common neighbour in the core, its diameter is 2. The edge set of G can be partitioned as  For illustration of Theorem 4.1, we consider two non-orientable prismatic graphs: rotator and twister. The rotator and twister are shown in 2 and 3, and their PI indices are 120 and 154, respectively.