The spectral profile of a graph is a natural generalization of the classical notion of its Rayleigh quotient. Roughly speaking, given a graph G, for each 0< δ < 1, the spectral profile Λ<sub>G</sub>(δ) minimizes the Rayleigh quotient (from the variational characterization) of the spectral gap of the Laplacian matrix of G over vectors withâ€¦Â (More)