Bounds for the positive or negative inertia index of a graph

TL;DR

This paper establishes bounds on the positive and negative inertia indices of a graph based on its matching and cyclomatic numbers, and characterizes graphs that reach these bounds.

Contribution

It provides new bounds for the inertia indices of graphs and characterizes the extremal graphs achieving these bounds.

Findings

Bounds for positive and negative inertia indices in terms of matching and cyclomatic numbers.

Characterization of graphs attaining the bounds.

Theoretical framework for inertia index analysis.

Abstract

Let $G$ be a graph and let $A(G)$ be adjacency matrix of $G$.The positive inertia index (respectively, the negative inertia index) of $G$, denoted by $p(G)$ (respectively, $n(G)$), is defined to be the number of positive eigenvalues (respectively, negative eigenvalues) of $A(G)$. In this paper, we present the bounds for $p(G)$ and $n(G)$ as follows: $$m(G)-c(G)\leq p(G)\leq m(G)+c(G), \ m(G)-c(G)\leq n(G)\leq m(G)+c(G),$$ where $m(G)$ and $c(G)$ are respectively the matching number and the cyclomatic number of $G$. Furthermore, we characterize the graphs which attain the upper bounds or the lower bounds respectively.