  The edge Szeged index of a graph $G$ is defined as
$Sz_{e}(G)=\sum\limits_{uv\in E(G)}m_{u}(uv|G)m_{v}(uv|G)$, 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.
