The inertia of weighted unicyclic graphs

TL;DR

This paper investigates the eigenvalue inertia of weighted unicyclic graphs, providing bounds and characterizations for graphs with specific eigenvalue distributions, advancing understanding of spectral properties in weighted graph theory.

Contribution

It establishes lower bounds for positive and negative eigenvalues and characterizes graphs attaining these bounds, focusing on weighted unicyclic graphs.

Findings

Lower bounds for positive and negative inertia indices are derived.

Characterizations of graphs with specific eigenvalue distributions are provided.

Results contribute to spectral graph theory and weighted graph analysis.

Abstract

Let $G_w$ be a weighted graph. The \textit{inertia} of $G_w$ is the triple $In(G_w)=\big(i_+(G_w),i_-(G_w), $ $ i_0(G_w)\big)$, where $i_+(G_w),i_-(G_w),i_0(G_w)$ are the number of the positive, negative and zero eigenvalues of the adjacency matrix $A(G_w)$ of $G_w$ including their multiplicities, respectively. $i_+(G_w)$, $i_-(G_w)$ is called the \textit{positive, negative index of inertia} of $G_w$, respectively. In this paper we present a lower bound for the positive, negative index of weighted unicyclic graphs of order $n$ with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound. Moreover, we characterize the weighted unicyclic graphs of order $n$ with two positive, two negative and at least $n-6$ zero eigenvalues, respectively.