Some useful lemmas on the edge Szeged index

The edge Szeged index of a graph $G$ is defined as $S{z}_e(G)=\sum _{uv\in E(G)}{m}_u(uv\mathrm{\mid }G)m_v(uv\mathrm{\mid }G)$, where $m_u(uv\mathrm{\mid }G)$ (resp., $m_v(uv\mathrm{\mid }G)$) 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.