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

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