Bicyclic graphs with maximal revised Szeged index

The revised Szeged index of a graph G is defined as Sz*(G) = Sigma(e=uv is an element of E) (n(u)(e) + n(0)(e)/2) (n(v)(e) + n(0)(e)/2), where n(u)(e) and n(v)(e) are, respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u, and n(0)(e) is the number of vertices equidistant to u and v. Hansen et al. used the AutoGraphiX and made the following conjecture about the revised Szeged index for a connected bicyclic graph G of order n >= 6: Sz*(G) <= {(n(3) + n(2) - n - 1)/4, if n is odd, (n(3) + n(2) - n)/4, if n is even. with equality if and only if G is the graph obtained from the cycle Cn-1 by duplicating a single vertex. This paper is to give a confirmative proof to this conjecture. (C) 2013 Elsevier B.V. All rights reserved.