A note on the positive semidefinitness of $A_α(G)$

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 $α\in\left[ 0,1\right]$, write $A_α\left( G\right)$ for the matrix $A_α\left( G\right) =αD\left( G\right) +(1-α)A\left( G\right) .$ Let $α_{0}\left( G\right)$ be the smallest $α$ for which $A_α(G)$ is positive semidefinite. It is known that $α_{0}\left( G\right) \leq1/2$. The main results of this paper are: (1) if $G$ is $d$-regular then $α_{0}=\frac{-λ_{\min}(A(G))}{d-λ_{\min}(A(G))},$ where $λ_{\min}(A(G))$ is the smallest eigenvalue of $A(G)$; (2) $G$ contains a bipartite component if and only if $α_{0}\left( G\right) =1/2$; (3) if $G$ is $r$-colorable, then $α_{0}\left( G\right) \geq1/r$.

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