On the characteristic polynomial and spectrum of Basilica Schreier graphs

Document Type : Workshop on Graphs, Topology and Topological Groups, Cape Town, South Africa

Authors

Dipartimento di Ingegneria, Universit`a degli Studi Niccol`o Cusano, Via Don Carlo Gnocchi, Roma, Italy

Abstract

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 $\{\Gamma_n\}_{n\geq 1}$, 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 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 $2$, and to determine how the eigenvalues are distributed with respect to such eigenvalue.

Keywords

Main Subjects


[1] L. Bartholdi and R. I. Grigorchuk, On the spectrum of Hecke type operators related to some fractal groups, translated from Tr. Mat. Inst. Steklova, 231 (2000), Din. Sist., Avtom. i Beskon. Gruppy, 5–45; Proc. Steklov Inst. Math., 231 (2000) no. 4 1–41.
[2] F. Belardo, S. Cioabă, J. Koolen and J. Wang, Open problems in the spectral theory of signed graphs, Art Discrete
Appl. Math., 1 (2018) no.2 pp. 23.
[3] F. Belardo and S. K. Simić, On the Laplacian coefficients of signed graphs, Linear Algebra Appl., 475 (2015) 94–113.
[4] I. Bondarenko, Groups generated by bounded automata and their Schreier graphs, Thesis (Ph.D.)–Texas A M Uni-
versity, ProQuest LLC, Ann Arbor, MI, 2007 pp. 162.
[5] I. Bondarenko and V. Nekrashevych, Post-critically finite self-similar groups, Algebra Discrete Math., 2 (2003) no.
4 21–32.
[6] A. Brzoska, C. George, S. Jarvis, L. G. Rogers and A. Teplyaev, Spectral properties of graphs associated to the
Basilica group, arXiv:1908.10505
[7] M. Cavaleri, D. D’Angeli and A. Donno, A group representation approach to balance of gain graphs, J. Algebraic
Combin., 54 (2021) no. 1 265–293.
[8] M. Cavaleri, D. D’Angeli, A. Donno and S. Hammer, Wiener, edge-Wiener, and vertex-edge-Wiener index of Basilica graphs, Discrete Appl. Math., 307 (2022) 32–49.
[9] M. Cavaleri, D. D’Angeli, A. Donno and E. Rodaro, On an uncountable family of graphs whose spectrum is a Cantor set, arXiv:2101.07547.
[10] T. Ceccherini-Silberstein, A. Donno, D. Iacono, The Tutte polynomial of the Schreier graphs of the Grigorchuk
group and the Basilica group, Ischia Group Theory 2010, 45–68, World Sci. Publ., Hackensack, N. J., 2012.
[11] D. M. Cvetković, M. Doob and H. Sachs, Spectra of graphs. Theory and application. Pure and Applied Mathematics, 87, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980 pp. 368.
[12] D. Cvetković, P. Rowlinson and S. Simić, An Introduction to the Theory of Graph Spectra, London Mathematical
Society Student Texts, 75, Cambridge University Press, Cambridge, 2010 pp. xii+364
[13] D. D’Angeli, A. Donno, M. Matter and T. Nagnibeda, Schreier graphs of the Basilica group, J. Mod. Dyn., 4 (2010)
no. 1 167–205.
[14] D. D’Angeli, A. Donno and T. Nagnibeda, Partition functions of the Ising model on some self-similar Schreier
graphs, Random walks, boundary and spectra, 277–304, Progr. Probab, 64, Birkhäuser/Springer Basel AG, Basel,
2011.
[15] R. Grigorchuk, T. Nagnibeda and A. Pérez, On spectra and spectral measures of Schreier and Cayley graphs,
arXiv:2007.03309
[16] R. Grigorchuk and Z. Šunić, Schreier spectrum of the Hanoi Towers group on three pegs, Analysis on graphs and its applications, 183–198, Proc. Sympos. Pure Math, 77, Amer. Math. Soc., Providence, RI, 2008.
[17] R. Grigorchuk and A. Żuk, On a torsion-free weakly branch group defined by a three-state automaton, Internat. J. Algebra Comput., 12 (2002) no. 1–2 223–246.
[18] R. Grigorchuk and A. Żuk, Spectral properties of a torsion-free weakly branch group defined by a three state
automaton. Computational and statistical group theory (Las Vegas, NV/Hoboken, NJ, 2001), 298 of Contemp.
Math., 57–82, Amer. Math. Soc., Providence, RI, 2002.
[19] J. L. Gross and T. W. Tucker, Generating all graph coverings by permutation voltage assignments, Discrete Math.,
18 (1977) no. 3 273–283.
[20] F. Harary, On the notion of balance of a signed graph, Michigan Math. J., 2 (1953–1954) 143–146 (1955).
[21] V. Nekrashevych, Self-similar Groups, Mathematical Surveys and Monographs, 117, American Mathematical Soci-ety, Providence, RI, (2005) pp. xii+231.
[22] L. G. Rogers and A. Teplyaev, Laplacians on the basilica Julia set, Commun. Pure Appl. Anal. 9 (2010) no. 1
211–231.
[23] T. Zaslavsky, A mathematical bibliography of signed and gain graphs and allied areas, Electron. J. Combin., 5
(1998) pp. 124.
Volume 11, Issue 3 - Serial Number 3
Introduction to the Proceedings of WGTTG2021
September 2022
Pages 153-179
  • Receive Date: 25 May 2021
  • Revise Date: 04 November 2021
  • Accept Date: 07 November 2021
  • Published Online: 01 September 2022