  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).
