Eigenvalue distribution in scale free graphs

Abstract

Scale free graphs can be found very often as models of real networks and are characterized by a power law degree distribution, that is, for a constant $\gamma\geq 1$ the number of vertices of degree $d$ is proportional to $d^{-\gamma}$. Experimental studies show that the eigenvalue distribution also follows a power law for the highest eigenvalues. Hence it has been conjectured that the power law of the degrees determines the power law of the eigenvalues. In this paper we show that we can construct a scale free graph with non highest eigenvalue power law distribution. For $\gamma=1$ we can construct a scale free graph with small spectrum and a regular graph with eigenvalue power law distribution.