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

Given a connected graph G, the edge Szeged index Sz(e)(G) is defined as Sz(e)(G) = Sigma e-uv is an element of E m(u)(e)m(v)(e), where m(u)(e) and m(v)(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. (C) 2018 Elsevier Inc. All rights reserved.