Some useful lemmas on the edge Szeged index

Shengjie He

paper

Some useful lemmas on the edge Szeged index

arXiv:1805.06578

Abstract

The edge Szeged index of a graph $G$ is defined as $S z_{e} (G) = uv \in E (G) \sum m_{u} (uv ∣ G) m_{v} (uv ∣ G)$ , where $m_{u} (uv ∣ G)$ (resp., $m_{v} (uv ∣ G)$ ) is the number of edges whose distance to vertex $u$ (resp., $v$ ) is smaller than the distance to vertex $v$ (resp., $u$ ), respectively. In this paper, we characterize the graph with minimum edge Szeged index among all the unicyclic graphs with given order and diameter.