The average scattering number of graphs

Abstract

The scattering number of a graph is a measure of the vulnerability of a graph. In this paper we investigate a refinement that involves the average of a local version of the parameter. If v is a vertex in a connected graph G , then s c v (G ) = max { ω (G − S v ) − | S v | }, where the maximum is taken over all disconnecting sets S v of G that contain v . The average scattering number of G denoted by s c av (G ), is defined as scav (G) = Σv ∈ V(G) scv(G) / n , where n will denote the number of vertices in graph G . Like the scattering number itself, this is a measure of the vulnerability of a graph, but it is more sensitive. Next, the relations between average scattering number and other parameters are determined. The average scattering number of some graph classes are obtained. Moreover, some results about the average scattering number of graphs obtained by graph operations are given.