We prove several types of Courant nodal domain theorems for generalized Laplacians on graphs, based on an invariant introduced by Urschel, which we call the "Urschel number", denoted ${\rm UN}({\bf f})$, of an eigenvector ${\bf f}$. We refine Urschel's invariant, and use perturbation techniques to obtain some new results. First, we show the existence of mutually orthogonal eigenvectors, such that if the $k$-th eigenvalue has multiplicity $m$, then for $0\le j\le m-1$, ${\rm UN}({\bf f}_{k+j})\le k+\min(j,(m-1)-j)$. Second, for a simple $k$-th eigenvalue, we classify the zeroes of ${\bf f}_k$ as either "shallow or "deep"; we obtain a number of results that say, roughly speaking, the more shallow vertices ${\bf f}_k$ has, the more control we have over our new invariants based on Urschel's. Our new invariants of an eigenvector, ${\bf f}_k$, are a sequence of integers whose minimum value is ${\rm UN}({\bf f}_k)$ and whose maximum, denoted ${\rm UN}_{\max{}}({\bf f}_k)$, is the maximum number of nodal domains of any possible positive/negative signing or "charge" of the zeroes of ${\bf f}_k$. An example of our second type of result is that if ${\bf f}_k$ has no deep vertices, then ${\rm UN}_{\max{}}({\bf f}_k)\le k$. We provide a number of examples to illustrate our main results, and how they differ from the situation in analysis. We also describe a minor improvement of the Gladwell-Zhu theorem for an orthonormal eigenbasis in the presence of eigenvalues of sufficient multiplicity.