On extremal cacti with respect to the edge Szeged index and edge-vertex Szeged index

The edge Szeged index and edge-vertex Szeged index of a graph are defined as $S{z}_e(G)=\sum _{uv\in E(G)}{m}_u(uv\mathrm{\mid }G)m_v(uv\mathrm{\mid }G)$ and $Sz_{ev}(G)=\frac{1}{2} \sum\limits_{uv \in E(G)}[n_{u}(uv|G)m_{v}(uv|G)+n_{v}(uv|G)m_{u}(uv|G)],$respectively,where$m_{u}(uv|G)$(resp\mathrm{.},$m_{v}(uv|G)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), and $n_u(uv\mathrm{\mid }G)$ (resp., $n_v(uv\mathrm{\mid }G)$) is the number of vertices whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), respectively. A cactus is a graph in which any two cycles have at most one common vertex. In this paper, the lower bounds of edge Szeged index and edge-vertex Szeged index for cacti with order $n$ and $k$ cycles are determined, and all the graphs that achieve the lower bounds are identified.