A note on the positive semidefinitness of $A_\alpha (G)$

TL;DR

This paper investigates the conditions under which the matrix $A_eta(G)$, a convex combination of the degree matrix and adjacency matrix of a graph, is positive semidefinite, providing explicit formulas and characterizations.

Contribution

It derives a formula for $eta_0(G)$ for regular graphs, characterizes bipartite graphs via $eta_0(G)$, and relates $eta_0(G)$ to the chromatic number.

Findings

$eta_0(G)$ equals $-rac{ ext{smallest eigenvalue of }A(G)}{d - ext{smallest eigenvalue}}$ for $d$-regular graphs.

$G$ has a bipartite component if and only if $eta_0(G)=1/2$.

For $r$-colorable graphs, $eta_0(G) ext{ is at least }1/r$.

Abstract

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $\alpha\in\left[ 0,1\right] $, write $A_{\alpha}\left( G\right) $ for the matrix \[ A_{\alpha}\left( G\right) =\alpha D\left( G\right) +(1-\alpha)A\left( G\right) . \] Let $\alpha_{0}\left( G\right) $ be the smallest $\alpha$ for which $A_{\alpha}(G)$ is positive semidefinite. It is known that $\alpha_{0}\left( G\right) \leq1/2$. The main results of this paper are: (1) if $G$ is $d$-regular then \[ \alpha_{0}=\frac{-\lambda_{\min}(A(G))}{d-\lambda_{\min}(A(G))}, \] where $\lambda_{\min}(A(G))$ is the smallest eigenvalue of $A(G)$; (2) $G$ contains a bipartite component if and only if $\alpha_{0}\left( G\right) =1/2$; (3) if $G$ is $r$-colorable, then $\alpha_{0}\left( G\right) \geq1/r$.