A connection between the Aα-spectrum and Lovász theta number

Abstract

We show that the smallest α so that $\alpha D+(1-\alpha )A\succcurlyeq 0$ is at least $1/\vartheta \left(\bar{G}\right)$, significantly improving upon a result due to Nikiforov and Rojo (2017). In fact, we display a stronger connection: if the nonzero entries of A are allowed to vary and those of D vary accordingly, then we show that this smallest α is in fact equal to $1/\vartheta \left(\bar{G}\right)$. We also show other results obtained as an application of this optimization framework, including a connection to the well-known quadratic formulation for $\omega \left(G\right)$ due to Motzkin and Straus (1964).