  The edge-connectivity matrix of a weighted graph is the matrix whose
off-diagonal $v$-$w$ entry is the weight of a minimum edge cut separating
vertices $v$ and $w$. Its computation is a classical topic of combinatorial
optimization since at least the seminal work of Gomory and Hu. In this article,
we investigate spectral properties of these matrices. In particular, we provide
tight bounds on the smallest eigenvalue and the energy. Moreover, we study the
eigenvector structure and show in which cases eigenvectors can be easily
obtained from matrix entries. These results in turn rely on a new
characterization of those nonnegative matrices that can actually occur as
edge-connectivity matrices.
