Rank of non-negative matrices

Are there some easy ways to verify conditions on a nonnegative matrix with $0$ diagonal which ensure that it is conditionally negative definite (i.e., $x^TAx le 0$ for all $x$ with $x^Tmathbf{1}=0$ and equality holds if and only if $x=0$)? Similarly for conditionally positive definite.

I am thinking of adjacency matrices on graphs; so, easy to verify condition means something that has an easy to understand interpretation in terms of the graphs, e.g., connectedness. What is a reference where this condition is stated explicitly? Thanks.