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

Abstract Given a connected graph G, the edge-Szeged index Sze(G) is defined as S z e ( G ) = ∑ e = u v ∈ E m u ( e ) m v ( 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.

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