Physica A 356 (2005) 1–5 Spectral properties of the Apollonian network R.F.S. Andrade à , J.G.V. Miranda Instituto de Fı´sica, Universidade Federal da Bahia, 40.130-240 Salvador, Brazil Available online 8 June 2005 Abstract A first characterization of the spectral properties of the Apollonian network is presented. Results are based on the numerically evaluated eigenvalue spectra of adjacency matrices for the first nine generations of the network. The main features refer to the presence of several gaps on the eigenvalue axis and a very large number of degenerated eigenvalues. The increase in the degeneracy of a family of eigenvalues is found to obey a very simple empirical relation. r 2005 Elsevier B.V. All rights reserved. Keywords: Complex networks; Eigenvalue spectrum; Apollonian network The recent investigation of several properties of networks [1,2] made it possible to characterize many aspects of systems with a very large number of components which interact with each other in a complex way. Analyses have included the characterization of topological aspects of the networks, the identification of certain classes with well-defined properties (random, small-world, scale-free, etc.), the evaluation of dynamical and physical properties of systems defined on these structures, the eigenvalue spectrum of the adjacency matrix (AM), and so on [3]. The AM indicates how the sites in a network are connected to each other containing, this way, the whole information about the network. The eigenvalue spectrum of the AM is a convenient way to compact information on a given system and also to compare distinct networks [3]. ARTICLE IN PRESS www.elsevier.com/locate/physa 0378-4371/$-see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.05.003 à Corresponding author. E-mail address: randrade@ufba.br (R.F.S. Andrade).