The Hosoya Polynomial of the Schreier Graphs of the Grigorchuk Group and the Basilica Group

The Grigorchuk group was first introduced by R. Grigorchuk in 1980. Also the Basilica group was introduced in 2002 by R. Grigorchuk and A. Zuk. In the following years, it was shown that these groups have deep connections with profinite group theory and complex dynamics. These groups have been proven to provide the self-similarity property, reflecting the fractalness of some limit objects associated with them. The Schreier graph codifies the intangible structure of a group. It establishes an equivalence relationship created by cosets. The Schreier graphs of the Grigorchuck group and the Basilica group are a combination of cycles arranged in a tree-like form due to the recursive expression of the generators of these groups. In this work, we study the Hosoya polynomial of these graphs and try to characterize them.

Keywords:

Basilica group, Grigorchuk group Hosoya polynomial, Schreier graph,

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