A connection between the $A_\alpha$-spectrum and the Lov\'asz theta number

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