  We initially consider a single-particle tight-binding model on the
Regularized Apollonian Network (RAN). The RAN is defined starting from a
tetrahedral structure with four nodes all connected (generation 0). At any
successive generations, new nodes are added and connected with the surrounding
three nodes. As a result, a power-law cumulative distribution of connectivity
$P(k)\propto {1}/{k^{\eta}}$ with $\eta=\ln(3)/\ln(2) \approx 1.585$ is
obtained.
  The eigenvalues of the Hamiltonian are exactly computed by a recursive
approach for any size of the network. In the infinite size limit, the density
of states and the cumulative distribution of states (integrated density of
states) are also exactly determined. The relevant scaling behavior of the
cumulative distribution close to the band bottom is shown to be power law with
an exponent depending on the spectral dimension and not on the embedding
dimension.
  We then consider a gas made by an infinite number of non-interacting bosons
each of them described by the tight-binding Hamiltonian on the RAN and we prove
that, for sufficiently large bosonic density and sufficiently small
temperature, a macroscopic fraction of the particles occupy the lowest
single-particle energy state forming the Bose-Einstein condensate. We determine
no only the transition temperature as a function of the bosonic density, but
also the fraction of condensed particle, the fugacity, the energy and the
specific heat for any temperature and bosonic density.
