The extremal unicyclic graphs with given diameter and minimum edge revised Szeged index

Abstract

<abstract><p>Let $ H $ be a connected graph. The edge revised Szeged index of $ H $ is defined as $ Sz^{\ast}_{e}(H) = \sum\limits_{e = uv\in E_H}(m_{u}(e|H)+\frac{m_{0}(e|H)}{2})(m_{v}(e|H)+\frac{m_{0}(e|H)}{2}) $, where $ m_{u}(e|H) $ (resp., $ m_{v}(e|H) $) is the number of edges whose distance to vertex $ u $ (resp., $ v $) is smaller than to vertex $ v $ (resp., $ u $), and $ m_{0}(e|H) $ is the number of edges equidistant from $ u $ and $ v $. In this paper, the extremal unicyclic graphs with given diameter and minimum edge revised Szeged index are characterized.</p></abstract>