  The thermodynamic properties of non interacting bosons on a complex network
can be strongly affected by topological inhomogeneities. The latter give rise
to anomalies in the density of states that can induce Bose-Einstein
condensation in low dimensional systems also in absence of external confining
potentials. The anomalies consist in energy regions composed of an infinite
number of states with vanishing weight in the thermodynamic limit. We present a
rigorous result providing the general conditions for the occurrence of
Bose-Einstein condensation on complex networks in presence of anomalous
spectral regions in the density of states. We present results on spectral
properties for a wide class of graphs where the theorem applies. We study in
detail an explicit geometrical realization, the comb lattice, which embodies
all the relevant features of this effect and which can be experimentally
implemented as an array of Josephson Junctions.
