On the spectral closeness and residual spectral closeness of graphs

Abstract

The spectral closeness of a graph G is defined as the spectral radius of the closeness matrix of G, whose (u, v)-entry for vertex u and vertex v is 2−dG(u,v) if u ≠ v and 0 otherwise, where dG(u, v) is the distance between u and v in G. The residual spectral closeness of a nontrivial graph G is defined as the minimum spectral closeness of the subgraphs of G with one vertex deleted. We propose local grafting operations that decrease or increase the spectral closeness and determine those graphs that uniquely minimize and/or maximize the spectral closeness in some families of graphs. We also discuss extremal properties of the residual spectral closeness.