On the edge-Szeged index of unicyclic graphs with given diameter

Given a connected graph G, the edge-Szeged index Sze(G) is defined as Sze(G)=∑e=uv∈Emu(e)mv(e), 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. In this paper, some extremal problems on the edge-Szeged index of unicyclic graphs are considered. All the n-vertex unicyclic graphs with a given diameter having the minimum edge-Szeged index are identified. Using a unified approach we identify the n-vertex unicyclic graphs with the minimum, second minimum, third minimum and fourth minimum edge-Szeged indices.