Bicyclic graphs with maximal edge revised Szeged index

The edge revised Szeged index Sze∗(G) is defined as Sze∗(G)=∑e=uv∈E(mu(e)+m0(e)/2)(mv(e)+m0(e)/2), where mu(e) and mv(e) are, respectively, the number of edges of G lying closer to vertex u than to vertex v and the number of edges of G lying closer to vertex v than to vertex u, and m0(e) is the number of edges equidistant to u and v. In this paper, we give an upper bound of the edge revised Szeged index for a connected bicyclic graphs with size m≥5, that is, Sze∗(G)≤{(m3−4m+16)/4,if  m  is odd ,(m3−4m+18)/4,if  m  is even with equality if and only if G is the graph obtained from the cycle Cm−2 by duplicating a single vertex.