Unicyclic graphs and the inertia of the distance squared matrix

TL;DR

This paper investigates the inertia of the distance squared matrix in unicyclic graphs, extending known results from trees and developing new tools to analyze how specific vertices influence matrix inertia.

Contribution

It introduces general tools for analyzing how pendant and degree-2 vertices affect the inertia of the distance squared matrix, extending existing results to unicyclic graphs.

Findings

Extended inertia results from trees to certain unicyclic graphs

Developed tools for analyzing vertex effects on matrix inertia

Provided an alternative proof of known inertia results

Abstract

A result of Bapat and Sivasubramanian gives the inertia of the distance squared matrix of a tree. We develop general tools on how pendant vertices and degree 2 vertices affect the inertia of the distance squared matrix and use these to give an alternative proof of this result. We further use these tools to extend this result to certain families of unicyclic graphs, and we explore how far these results can be extended.