On the characteristic polynomial and spectrum of Basilica Schreier graphs

Cavaleri, Matteo; D&#039;Angeli, Daniele; Donno, Alfredo

doi:10.22108/toc.2021.128870.1858

تعداد نشریات	43
تعداد شماره‌ها	1,724
تعداد مقالات	14,106
تعداد مشاهده مقاله	34,324,981
تعداد دریافت فایل اصل مقاله	13,753,862

	On the characteristic polynomial and spectrum of Basilica Schreier graphs
Transactions on Combinatorics
مقاله 4، دوره 11، شماره 3، آذر 2022، صفحه 153-179 اصل مقاله (931.88 K)
نوع مقاله: Workshop on Graphs, Topology and Topological Groups, Cape Town, South Africa
شناسه دیجیتال (DOI): 10.22108/toc.2021.128870.1858
نویسندگان
Matteo Cavaleri؛ Daniele D'Angeli؛ Alfredo Donno^*
Dipartimento di Ingegneria, Universit`a degli Studi Niccol`o Cusano, Via Don Carlo Gnocchi, Roma, Italy
چکیده
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.
کلیدواژه‌ها
Basilica Schreier graph؛ Characteristic polynomial؛ Signed graph؛ Coefficient Theorem؛ Basic figure

مراجع
[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.
آمار تعداد مشاهده مقاله: 312 تعداد دریافت فایل اصل مقاله: 453

سامانه مدیریت نشریات علمی. طراحی و پیاده سازی از سیناوب

پیوندهای مفید

اخبار و اعلانات

آمار

On the characteristic polynomial and spectrum of Basilica Schreier graphs