ON THE CHARACTERISTIC POLYNOMIAL AND SPECTRUM OF BASILICA SCHREIER GRAPHS

The basilica group is one of the most studied automaton groups, and many papers havebeen devoted to the investigation of the characteristic polynomial and spectrum of the associatedschreier graphs {γ𝑛}𝑛≥1, 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 coeffcient theorem forsigned graphs. we introduce a signed version γ𝑛 of the basilica schreier graph γ𝑛, 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 γ𝑛 into three parts, using the self-similarity of γ𝑛, via a detailed investigation of its basic figures. by gluing together these relations, we nd out a new recursive equation which expresses the characteristic polynomial of γ𝑛 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 2, and to determine how the eigenvalues are distributed with respect to such eigenvalue.