Resolving share and topological index

Abstract

A vertex v of a graph G uniquely determines (resolves) a pair (v1,v2) of vertices of G if the distance between v and v\ is different from the distance between v and V2. In this paper, we quantify the involvement of each vertex v of G in uniquely determining (resolving) a pair (v1, V2) of vertices of G, which is called the resolving share of v for the pair (v1, v2). Using this quantity, we define a distance-based topological index of a graph, which reflects the topology of that graph according to the resolvability behavior of each of its vertices, and is called the resolving topological index. Then, we compute the resolving topological index of several graphs. © 2018 Utilitas Mathematica Publishing Inc. All rights reserved.