Bounds on the edge-Wiener index of cacti with $n$ vertices and $t$ cycles

TL;DR

This paper establishes bounds on the edge-Wiener index for cacti graphs with a fixed number of vertices and cycles, identifying extremal structures within this class.

Contribution

It provides the first known bounds on the edge-Wiener index for cacti with given vertices and cycles, and characterizes the extremal graphs achieving these bounds.

Findings

Derived upper and lower bounds for the edge-Wiener index.

Characterized extremal cacti graphs that attain these bounds.

Enhanced understanding of graph distance measures in cactus structures.

Abstract

The edge-Wiener index $W_e(G)$ of a connected graph $G$ is the sum of distances between all pairs of edges of $G$. A connected graph $G$ is said to be a cactus if each of its blocks is either a cycle or an edge. Let $\mathcal{G}_{n,t}$ denote the class of all cacti with $n$ vertices and $t$ cycles. In this paper, the upper bound and lower bound on the edge-Wiener index of graphs in $\mathcal{G}_{n,t}$ are identified and the corresponding extremal graphs are characterized.