Deterministic small-world graphs and the eigenvalue power law of Internet

Many relevant real-life networks like the WWW, Internet, transportation and communication networks, or even biological and social networks can be modelled by small-world scale-free graphs. These graphs have strong local clustering (vertices have many mutual neighbors), a small diameter and a distribution of degrees according to a power law. On the other hand, the knowledge of the spectrum of a graph is important for the relation which the eigenvalues and their multiplicities have with relevant graph invariants and topological and communication properties such as diameter, bisection width, distances, connectivity, expansion, partitions, edge-loading distribution etc. In this paper we introduce a new family of deterministic small-world graphs, we determine analytically their spectra and we show how these graphs can model the eigenvalue power-law of the Internet network.