Wiener Index of Graphs with Radius Two

Abstract

The Wiener index of a graph is the sum of the distances between all pairs of vertices. It has been one of main descriptors that correlate a chemical compound's molecular graph with experimentally gathered data regarding the compound's characteristics. We characterize graphs with the maximum Wiener index among all graphs of order . with radius two. In addition, we pose a conjecture concerning the minimum Wiener index of graphs with given radius. If this conjecture is true, it will be able to answer an open question by You and Liu (2011).

Mathematical aspects of Wiener index

“…Regarding this problem, Chen et al [5] stated the following conjecture. For integers n, r, and s with n ≥ 2r, r ≥ 3, and n − 2r + 1 ≥ s ≥ 1, construct a graph G n,r,s from a 2r-cycle v 1 v 2 · · · v 2r so that v 1 is replaced by K s and v 2 is replaced by K n−2r+2−s , connect v 2r to each vertex of K s , connect each vertex of K s to each vertex of K n−2r+2−s , and connect each vertex of K n−2r+2−s to v 3 (in other words v 1 is replicated s−1 times, and v 2 is replicated n−2r +1−s times).…”

Extremal total distance of graphs of given radius I

“…The question of determining the minimum total distance among all graphs or digraphs of order and (out)radius and characterizing the extremal (di)graphs has been solved for large enough compared with . This partially solves the conjecture of Chen et al [4]. It might be very challenging to solve the question completely for every order, as there might be sporadic extremal graphs other than which are also different from those conjectured in the initial conjecture.…”

“…The question of determining the minimum total distance among all graphs or digraphs of order n and (out)radius r and characterizing the extremal (di)graphs has been solved for n large enough compared with r. This partially solves the conjecture of Chen, Wu and An [4].…”

“…The extremal graphs attaining the minimal total distance among all graphs with order and radius are easily characterized; complete graphs when , complete graphs minus a maximum matching when and and complete graphs minus a maximum matching and an additional edge adjacent to the vertex not in the maximum matching, when and . For the question is harder and a conjecture of the extremal graphs was made by Chen et al [4]. Here is a cycle in which we take blow‐ups in two consecutive vertices by cliques and , as defined in Section 2.…”

Maximum size of digraphs of given radius

“…Here there are 6 non-isomorphic extremal graphs (a = 1 gives twice the same graph). Chen, Wu and An [6] had conjectured that the graphs G n,r,s minimize the total distance among all graphs of order n and radius r. This was proven for n sufficiently large compared with r in [4]. It is natural to think there is a relation between maximizing the size and minimizing the total distance.…”