paper

On extremal cacti with respect to the edge revised Szeged index

arXiv:1804.06009

Abstract

Let $G$ be a connected graph. The edge revised Szeged index of $G$ is defined as $S z_{e}^{*} (G) = e = uv \in E (G) \sum (m_{u} (e ∣ G) + \frac{m _{0} ( e ∣ G )}{2}) (m_{v} (e ∣ G) + \frac{m _{0} ( e ∣ G )}{2})$ , where $m_{u} (e ∣ G)$ (resp., $m_{v} (e ∣ G)$ ) is the number of edges whose distance to vertex $u$ (resp., $v$ ) is smaller than the distance to vertex $v$ (resp., $u$ ), and $m_{0} (e ∣ G)$ is the number of edges equidistant from both ends of $e$ . In this paper, we give the minimal and the second minimal edge revised Szeged index of cacti with order $n$ and $k$ cycles, and all the graphs that achieve the minimal and second minimal edge revised Szeged index are identified.