On the eccentric distance sum of unicyclic graphs with a given matching number

TL;DR

This paper characterizes unicyclic graphs with a fixed number of vertices and matching number that minimize their eccentric distance sum, providing insights into their structural properties.

Contribution

It identifies unicyclic graphs with given matching number that have the smallest and second smallest eccentric distance sums.

Findings

Unicyclic graphs with minimal eccentric distance sum are characterized.

The second minimal eccentric distance sum unicyclic graphs are also characterized.

Results contribute to understanding the relationship between graph structure and eccentric distance sums.

Abstract

Let $G = (V_G,E_G)$ be a simple connected graph. The eccentric distance sum of $G$ is defined as $\xi^d(G)=\sum_{v \in V_G}\,\varepsilon_G(v)D_G(v),$ where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$ and $D_G(v)=\sum_{u \in V_G}\,d(u,v)$ is the sum of all distances from the vertex $v$. In this paper, we characterize $n$-vertex unicyclic graphs with given matching number having the minimal and second minimal eccentric distance sums, respectively.