Graphs with zero spectrum / nilpotent symmetric matrices

Question

Is there a graph theoretic characterization of those graphs with zero spectrum?

Alternatively, can one at least characterize all symmetric nilpotent (complex) matrices, so that one could recognize those graphs by their adjacency matrices?

Thank you in advance!

Answer 1

(Assuming simple, undirected graphs) The $k$-th spectral moment is defined as $s_k=\lambda_1^k+...+\lambda_n^k$, where $\{\lambda_i\}$ are the eigenvalues. The $k$-th moment has an interpretation: it is the number of closed paths of length $k$ (these paths can be self-intersecting). If all eigenvalues are 0, ...

Answer 2

Observe that that the trace of $A^2$ is just the number of non zero entries of A, but this trace is zero.