Consider the Laplacian $L$ of a bipartite graph. Is there any generic understanding we have about what $1/(z-L)$ looks like? [say $z > \lambda_\max(L)$)]
You can consider variations of $L$ like matrices which have their non-zero entries exactly where a graph Laplacian would have them but may be the off-diagonal entries are not restricted to $0/1$ but are may be $0/1/{-1}$.