-1Department of Mathematics, G.B Pant Government Engineering College, New Delhi, India

-2,3Department of Mathematics, SSN Engineering College, Chennai, India

Let G be a finite, simple, undirected and connected graph with vertex set V(G). For an ordered subset W={w_1,w_2,…w_k} of vertices and a vertex 'v' in G, the representation of 'v' with respect to W is the ordered k-tuple r(v|W)=(d(v,w_1),d(v,w_2),⋯,d(v,w_k)), where d(x,y) is the distance between the vertices 'x' and 'y'. The set W is a resolving set (or locating set) for G if every two vertices of G have distinct representations . The metric dimension of G (dim(G)) is the minimum cardinality of a resolving set for G. The resolving set containing minimum number of vertices is called a basis (or reference set) for G. In this paper, we introduce a new resolving parameter, average resolving number (r_av (G)) of G and a new graph, namely resolving excellent graph, both depends on minimum cardinality of the resolving set of the graph G containing each element of V(G). The average resolving number of some standard graphs, product graphs, m-bismuth chain and m-lead chloride chain are found and we investigate the resolving excellent graphs. We find the upper Bounds of average resolving number in terms of order(n) and diameter(d) of the connected graph G. The need for this new resolving parameter (r_av (G)) is explained using the isomers of alkanes, which are chemical compounds having same molecular formula but different chemical structures.

Metric dimension, Resolving Set, Resolving Number, Average Resolving Number, Resolving Excellent Graph, Bismuth Tri-Iodide, Lead Chloride