Bicyclic graphs with maximal revised Szeged index

The revised Szeged index of a graph GG is defined as Sz∗(G)=∑e=uv∈E(nu(e)+n0(e)/2)(nv(e)+n0(e)/2)Sz∗(G)=∑e=uv∈E(nu(e)+n0(e)/2)(nv(e)+n0(e)/2), where nu(e)nu(e) and nv(e)nv(e) are, respectively, the number of vertices of GG lying closer to vertex uu than to vertex vv and the number of vertices of GG lying closer to vertex vv than to vertex uu, and n0(e)n0(e) is the number of vertices equidistant to uu and vv. Hansen et al. used the AutoGraphiX and made the following conjecture about the revised Szeged index for a connected bicyclic graph GG of order n≥6n≥6: Sz∗(G)≤{(n3+n2−n−1)/4,if  n  is odd ,(n3+n2−n)/4,if  n  is even . with equality if and only if GG is the graph obtained from the cycle Cn−1Cn−1 by duplicating a single vertex. This paper is to give a confirmative proof to this conjecture.