On extremal cacti with respect to the edge revised Szeged index

TL;DR

This paper characterizes extremal cacti graphs with respect to the edge revised Szeged index, identifying those with minimal and second minimal index values among cacti with given order and number of cycles.

Contribution

It determines the extremal cacti graphs with the lowest and second-lowest edge revised Szeged index for specified parameters, expanding understanding of graph invariants.

Findings

Identified cacti with minimal edge revised Szeged index.

Determined second minimal index cacti.

Characterized all extremal graphs achieving these indices.

Abstract

Let $G$ be a connected graph. The edge revised Szeged index of $G$ is defined as $Sz^{\ast}_{e}(G)=\sum\limits_{e=uv\in E(G)}(m_{u}(e|G)+\frac{m_{0}(e|G)}{2})(m_{v}(e|G)+\frac{m_{0}(e|G)}{2})$, 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.