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Wiener Index of Chain Graphs SHAHISTHA, K ARATHI BHAT and SUDHAKARA G IAENG International Journal of Applied Mathematics, 50:4, IJAM_50_4_10 ______________________________________________________________________________________ Wiener Index of Chain Graphs SHAHISTHA, K ARATHI BHAT and SUDHAKARA G Abstract—A bipartite graph is called a chain graph if the concepts and notations. neighborhoods of the vertices in each partite set form a chain The distance d(u; v) between a pair of vertices u; v in G is with respect to set inclusion. Chain graphs are discovered the length of the shortest path between u and v if exists, and re-discovered by various researchers in various contexts. Also, they were named differently according to the applications else d(u; v) = 1. The Wiener index is a graph invariant in which they arise. In the field of Spectral Graph Theory, which belongs to the molecular structure descriptors known chain graphs play a remarkable role. They are characterized as topological indices, which are used for the designing the as graphs with the largest spectral radius among all the molecules with required properties according to the practical connected bipartite graphs with prescribed number of edges convinience. The Wiener index w(G) of a graph G is the and vertices. Even though chain graphs are significant in the field of Spectral Graph Theory, the area of graph parameters sum of all distances between all pairs of vertices in G. remains untouched. Wiener index is one of the oldest and most X studied topological indices, both from theoretical point of view w(G) = d(u; v): and applications. The Wiener index of a graph is defined as fu;vg2V (G) the sum of distances between every pair of vertices in it. In this article, we give bounds for Wiener index and hence for The line graph L(G) of a graph G is a graph whose vertices average distance of chain graphs. We also present a quadratic are the edges of G, with two vertices of L(G) are adjacent time algorithm that returns a realising chain graph G (if exists) whenever the corresponding edges in G share a vertex in whose Wiener index is the given integer w. The algorithm presented in this article contributes to the existing knowledge common. The edge Wiener index, we(G) is defined as the in the theory of inverse Wiener index problem. We conclude sum of distances between all pairs of edges of the underlying this article by exploring some more graph parameters called (connected) graph. In other words, the edge Wiener index edge Wiener index, hyper Wiener index and Zagreb indices of is defined as the sum of distances of their corresponding chain graphs. vertices in the line graph L(G) [2]. That is, Index Terms—Bipartite graphs, Bi-star graph, Edge Wiener index, Average distance, Zagreb index, Hyper Wiener index. we(G) = w(L(G)): Due to the significance of Wiener index in the area of molecular chemistry, the inverse Wiener index problem is I. INTRODUCTION posed and some classes have been investigated in [3] and BIGRAPH or a bipartite graph is a graph G whose [4]. Peterson graph is one of the significant cubic graph A vertex set V (G) can be partitioned into two subsets which serves as the example/counter example for most of V1 and V2 such that every edge of G joins a vertex of various problems in graph theory. Wiener index and other V1 with a vertex of V2. We denote a bipartite graph with chemical indices of generalized Peterson’s graph has been the bipartition V (G) = V1 [ V2 by G(V1 [ V2;E). If G investigated in detail by the authors of [5]. The average path contains every edge joining the vertices of V1 and V2, length Av(G) of a graph G is another distance related graph then it is a complete bipartite graph. A complete bipartite parameter which is defined as follows: graph with jV j = m; jV j = n is denoted by K . A star 1 2 m;n P d(u; v) graph is a complete bipartite graph where at least one of fu;vg2V (G) jVij = 1(i = 1; 2). A bi-star graph B(p; q) is the graph Av(G) = n : obtained by joining the central (apex) vertices of two star 2 graphs K1;p−1 and K1;q−1 by an edge. We write u ∼ v or This parameter is one of the quantitative performance mea- u ∼G v if the vertices u and v are adjacent in G, u v sure to describe and compare two or more graphs when it is or u G v if they are not. The neighborhood of a vertex considered as a network. The Zagreb indices are another set u 2 V (G) is the set NG(u) consisting of all the vertices v of parameters, that can be viewed as structure descriptors of such that v ∼ u in G. For a bipartite graph, the adjacency the underlying graph of the molecule. In 1972, the Zagreb ! 0 B indices have been introduced to explain mathematically, the matrix can be written as , where B is called the BT 0 properties of compounds at the molecular level. The first Zagreb index and the second Zagreb index are denoted by biadjacency matrix. A vertex v 2 Vi (i = 1; 2) in a bipartite M1(G) and M2(G) respectively. That is graph G(V1 [ V2;E) is said to be dominating vertex if N (v) = V j 6= i v X X G j where . In other words, is of full degree M (G) = deg(v)2 = [deg(u) + deg(v)] with respect to the other partite set. We refer [1] for further 1 v2V (G) (u;v)2E(G) Manuscript received June 18, 2020; revised September 22, 2020. and X Department of Mathematics, Manipal Institute of Technology, M (G) = deg(u)deg(v): Manipal Academy of Higher Education, Manipal, 576104, India. 2 email: ([email protected], [email protected], (u;v)2E(G) [email protected]). Corresponding author: K ARATHI BHAT, Manipal Institute of Technol- These indices reflect the extent of branching of the molecular ogy, Manipal Academy of Higher Education, Manipal. carbon atom skeleton [6], [7]. Volume 50, Issue 4: December 2020 IAENG International Journal of Applied Mathematics, 50:4, IJAM_50_4_10 ______________________________________________________________________________________ A. Chain graphs NG(v10) ⊆ NG(v9) ⊆ NG(v8) ⊆ NG(v7) ⊆ NG(v6): The vertices v and v are dominating in the above example. A class of sets S = fS1;S2; :::; Sng is called a chain 1 6 with respect to the operation of set inclusion if for every Chain graphs are the maximizers for the largest eigenvalue of the graphs among bipartite graphs (connected) of fixed Si;Sj 2 S, either Si ⊆ Sj or Sj ⊆ Si. Chain graphs are the special case of bipartite graphs, which is defined as follows. size and order. It is interesting to note that chain graphs have 1 Definition 1.1: A graph is called a chain graph if it is no eigenvalues in the interval (0; 2 ) and all their non-zero bipartite and the neighborhoods of the vertices in each partite eigenvalues are simple [8]. It was conjectured that no chain set form a chain with respect to set inclusion. graph shares a non-zero eigenvalue with its vertex deleted In other words, for every two vertices u and v in the same subgraphs. But later it was disproved and a weaker version of the conjecture came out to be true. That is, the conjecture partite set and their neighborhoods NG(u) and NG(v), is true only for the subgraphs obtained by deleting vertices either NG(u) ⊆ NG(v) or NG(v) ⊆ NG(u). Chain graphs are often called Optimal graphs, Double Nested graphs of maximum degrees (dominating vertices) in either of the (DNG) or Difference graphs [8], [9], [10]. The color classes color classes [8]. For the other important and interesting results concerned with the chain graphs, the readers are of a chain graph G(V1 [ V2;E) can be partitioned into h non-empty cells given by referred to [9], [10], [11], [12], [13], [14], [15], [16] and [17]. The outline of the remainder of this paper is as follows: V1 = V11 [ V12 [ ::: [ V1h In section 2, after the introduction, we give bounds as well and as some characterizations for Wiener index of chain graphs. V2 = V21 [ V22 [ ::: [ V2h In the third section, for a given pair of integers (w; n), we present an algorithm which returns the chain graph of order such that N (u) = V [V [:::[V , for any vertex G 21 22 2(h−i+1) n and Wiener index w if exists. Section 4 deals with ‘edge u 2 V , 1 ≤ i ≤ h. If m = jV j and n = jV j, then we 1i i 1i i 2i version’ of Wiener index of chain graphs. We conclude this write article by giving bounds for Zagreb indices of chain graphs. G = DNG(m1; m2; :::; mh; n1; n2; :::; nh): II. WIENER INDEX If mi = ni = 1 for all 1 ≤ i ≤ h, then the graph is called half graph [8]. We note that, every partite set in a chain graph In this section, we give an expression as well as bounds has at least one dominating vertex, that is, a vertex adjacent for Wiener index of chain graphs. to all the vertices of the other partite set. Chain graphs are Theorem 2.1: Let G(V1 [ V2;E) be a chain graph of size also known as Difference graphs due to the property that m where jV1j = p; jV2j = q. Then the Wiener index w(G) every vertex of vi of G can be assigned a real number ai of G is given by for which there exists a positive real number R such that w(G) = p2 + q2 + 3pq − p − q − 2m: jaij < R for all i and two vertices vi; vj are adjacent if and ja − a j ≥ R only if i j [8].
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