Edge-connectivity matrices and their spectra

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