On the characteristic polynomial and spectrum of Basilica Schreier graphs

The Basilica group is one of the most studied automaton groups, and many papers have been devoted to the investigation of the characteristic polynomial and spectrum of the associated Schreier graphs \{\Gamma_n\}_{n\geq ۱}, even if an explicit description of them has not been given yet.   Our approach to this issue is original, and it is based on the use of the Coefficient Theorem for signed graphs. We introduce a signed version \Gamma_n^- of the Basilica Schreier graph \Gamma_n, and we prove that there exist two fundamental relations between the characteristic polynomials of the signed and unsigned versions. The first relation comes from the cover theory of signed graphs. The second relation is obtained by providing a suitable decomposition of \Gamma_n into three parts, using the self-similarity of \Gamma_n, via a detailed investigation of its basic figures. By gluing together these relations, we find out a new recursive equation which expresses the characteristic polynomial of \Gamma_n as a function of the characteristic polynomials of the three previous levels. We are also able to give an explicit description of the eigenspace associated with the eigenvalue ۲, and to determine how the eigenvalues are distributed with respect to such eigenvalue.

کلیدواژه ها:

Basilica Schreier graph ، Characteristic polynomial ، signed graph ، Coefficient Theorem ، Basic figure