Bounds and Relations Involving Adjusted Centrality of the Vertices of a Tree

Abstract

Edge betweenness is a measure of the influence of edges in networks. The edge betweenness of an edge is defined to be the number of shortest paths between pairs of vertices that run along it. For a given graph $$G=(V(G),E(G))$$G=(V(G),E(G)), the adjusted betweenness centrality $$c_v^*$$cv? of a vertex $$v\in V(G)$$v?V(G) is defined to be the sum of the edge betweenness of all the edges incident to $$v$$v. In this paper we study the mathematical properties of $$c_v^*$$cv? for vertices in trees. We present lower and upper bounds for $$c_v^*$$cv? in terms of the number of vertices, vertex degree, eccentricity, transmission, maximum degree, diameter, and radius.