The extremal unicyclic graphs of the revised edge Szeged index with given diameter

Let $G$ be a connected graph. The revised edge Szeged index of $G$ is defined as $S{z}_e^{\ast }(G)=\sum _{e=uv\in E(G)}(m_u(e\mathrm{\mid }G)+\frac{m_0(e\mathrm{\mid }G)}{2})(m_v(e\mathrm{\mid }G)+\frac{m_0(e\mathrm{\mid }G)}{2})$, where $m_u(e\mathrm{\mid }G)$ (resp., $m_v(e\mathrm{\mid }G)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), and $m_0(e\mathrm{\mid }G)$ is the number of edges equidistant from both ends of $e$, respectively. In this paper, the graphs with minimum revised edge Szeged index among all the unicyclic graphs with given diameter are characterized.