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We study the mathematical aspects of the Bose Einstein Condensation for thepure hopping model describing arrays of Josephson junctions on non homogeneousnetworks. The graphs under investigation are obtained by adding density zeroperturbations to the homogeneous Cayley Trees. The resulting topological modelis described by the (opposite of the) adjacency operator on the graph. In thepresent paper we investigate some relevant spectral properties of the adjacencyof the perturbed network, such the appearance of the hidden spectrum below thenorm, the transience character and the Perron Frobenius distribution. All thementioned properties have a mathematical interest in itself, and have naturalapplications for the investigation of the BEC of Bardeen Cooper Bosons on thenetworks under consideration.

Harmonic analysis on perturbed Cayley Trees