Regarding interpretations of eigenvalues, I recall that the spectra of a graph can tell you if it is bipartite. I think a connected graph is bipartite if and only if $-\lambda$ is one of it's eigenvalues (here $\lambda$ is the largest eigenvalue). Please correct me if I've muddled the theorem.

Regarding interpretations of eigenvalues, I recall that the spectra of a graph can tell you if it is bipartite. I think a connected graph is bipartite if and only if $-\lambda$ is one of its eigenvalues (here $\lambda$ is the largest eigenvalue). Please correct me if I've muddled the theorem.