On the sharp bounds of bicyclic graphs regarding edge Szeged index

Abstract For a given graph G, its edge Szeged index is denoted by S z e ( G ) = ∑ e = u v ∈ E ( G ) m u ( e ) m v ( e ) , where mu(e) and mv(e) are the number of edges in G with distance to u less than to v and the number of edges in G further (distance) to u than v, respectively. In the paper, the bounds of edge Szeged index on bicyclic graphs are determined. Furthermore, the graphs that achieve the bounds are completely characterized.