Setup: Let $D = D^T > 0$ be a positive definite and diagonal $n\times n$ matrix, and let $A = A^T \in \mathbb{R}^{n\times n}$ be nonnegative with zero diagonals. That is, $a_{ij} \geq 0$ for $i\neq j$ and $a_{ii} = 0$. In my particular case, $A$ is an adjacency matrix of an undirected graph.
Question: Give necessary and sufficient conditions under which $D - A$ positive definite. Failing that, what's a necessary condition so I can understand the possible gap between necessity and sufficiency.
Partial Answer: A sufficient, but I believe not necessary, condition is that $\min_i D_{ii} > \rho(A) = \lambda_{\rm max}(A)$, where the last equality follows from the Perron theorem.
From the Disc Theorem, another sufficient condition is that $D_{ii} \geq \sum_{j=1}^n a_{ij}$ with strict inequality in at least one row.
Happy Thinking, -John