Some useful lemmas on the edge Szeged index

TL;DR

This paper investigates the edge Szeged index, a graph invariant related to distances between edges and vertices, and characterizes the unicyclic graphs with the smallest index given specific order and diameter constraints.

Contribution

It provides a characterization of the unicyclic graphs that minimize the edge Szeged index for fixed order and diameter.

Findings

Identifies the unicyclic graphs with minimum edge Szeged index.

Provides a mathematical characterization based on graph order and diameter.

Contributes to understanding of distance-based graph invariants.

Abstract

The edge Szeged index of a graph $G$ is defined as $Sz_{e}(G)=\sum\limits_{uv\in E(G)}m_{u}(uv|G)m_{v}(uv|G)$, where $m_{u}(uv|G)$ (resp., $m_{v}(uv|G)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), respectively. In this paper, we characterize the graph with minimum edge Szeged index among all the unicyclic graphs with given order and diameter.