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

Abstract

The edge Szeged index and edge-vertex Szeged index of a graph are defined as $Sz_{e}(G)=\sum\limits_{uv\in E(G)}m_{u}(uv|G)m_{v}(uv|G)$
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., $m_{v}(uv|G)$) and $n_{u}(uv|G)$ (resp., $n_{v}(uv|G)$) are the number of edges and vertices whose distances to vertex $u$ (resp., $v$) are smaller than the distances 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.