On graphs with exactly two positive eigenvalues

TL;DR

This paper characterizes all graphs with exactly two positive eigenvalues and one zero eigenvalue by introducing specific transformations that preserve inertia properties.

Contribution

It introduces congruent transformations for graphs and completely classifies graphs with two positive and one zero eigenvalues.

Findings

Identifies all graphs with exactly two positive eigenvalues and one zero eigenvalue.

Develops congruent transformations that preserve positive and negative inertia indices.

Provides a complete characterization of such graphs.

Abstract

The inertia of a graph $G$ is defined to be the triplet $In(G) = (p(G), n(G), $ $\eta(G))$, where $p(G)$, $n(G)$ and $\eta(G)$ are the numbers of positive, negative and zero eigenvalues (including multiplicities) of the adjacency matrix $A(G)$, respectively. Traditionally $p(G)$ (resp. $n(G)$) is called the positive (resp. negative) inertia index of $G$. In this paper, we introduce three types of congruent transformations for graphs that keep the positive inertia index and negative inertia index. By using these congruent transformations, we determine all graphs with exactly two positive eigenvalues and one zero eigenvalue.