Diameters, distortion, and eigenvalues

We study the relation between the diameter, the first positive eigenvalue of the discrete $p$-Laplacian and the $\ell_p$-distortion of a finite graph. We prove an inequality relating these three quantities and apply it to families of Cayley and Schreier graphs. We also show that the $\ell_p$-distortion of Pascal graphs, approximating the Sierpinski gasket, is bounded, which allows to obtain estimates for the convergence to zero of the spectral gap as an application of the main result.

[1]  R. Grigorchuk,et al.  Asymptotic aspects of Schreier graphs and Hanoi Towers groups , 2006 .

[2]  Zoran Sunic,et al.  The Spectral Problem, Substitutions and Iterated Monodromy , 2007 .

[3]  N. Alon,et al.  il , , lsoperimetric Inequalities for Graphs , and Superconcentrators , 1985 .

[4]  Andrzej Żuk,et al.  The Lamplighter Group as a Group Generated by a 2-state Automaton, and its Spectrum , 2001 .

[5]  N. Abreu.  Old and new results on algebraic connectivity of graphs , 2007 .

[6]  I. Bondarenko.  Groups generated by bounded automata and their schreier graphs , 2007 .

[7]  Mario Szegedy,et al.  In How Many Steps the k Peg Version of the Towers of Hanoi Game Can Be Solved? , 1999, STACS.

[8]  Nathan Linial,et al.  Girth and euclidean distortion , 2002, STOC '02.

[9]  R. Grigorchuk,et al.  Bernside's problem on periodic groups , 1980 .

[10]  T. Nagnibeda,et al.  Schreier graphs of the Basilica group , 2010 .

[11]  Matthias Hein,et al.  Spectral clustering based on the graph p-Laplacian , 2009, ICML '09.

[12]  Laurent Bartholdi,et al.  On the Spectrum of Hecke Type Operators related to some Fractal Groups , 1999 .

[13]  Jiri Matousek,et al.  Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[14]  Vadim A. Kaimanovich.  "münchhausen Trick" and Amenability of Self-similar Groups , 2005, Int. J. Algebra Comput..

[15]  Berndt Farwer,et al.  ω-automata , 2002 .

[16]  Rostislav I. Grigorchuk,et al.  On a Torsion-Free Weakly Branch Group Defined by a Three State Automaton , 2002, Int. J. Algebra Comput..

[17]  I. Bárány.  LECTURES ON DISCRETE GEOMETRY (Graduate Texts in Mathematics 212) , 2003 .

[18]  James Tanton.  Mathematics Galore!: The Tower of Hanoi , 2012 .

[19]  R. I. Grigorchuk,et al.  Just Infinite Branch Groups , 2000 .

[20]  Fan Chung Graham,et al.  The Diameter and Laplacian Eigenvalues of Directed Graphs , 2006, Electron. J. Comb..

[21]  R. Grigorchuk,et al.  From fractal groups to fractal sets , 2002, math/0202001.

[22]  J. Bourgain.  On lipschitz embedding of finite metric spaces in Hilbert space , 1985 .

[23]  Z. Sunic,et al.  Schreier spectrum of the Hanoi Towers group on three pegs , 2007, 0711.0068.

[24]  Hiroshi Takeuchi.  The spectrum of the $p$-Laplacian and $p$-harmonic morphisms on graphs , 2003 .

[25]  Josef Leydold,et al.  LARGEST EIGENVALUES OF THE DISCRETE P-LAPLACIAN OF TREES WITH DEGREE SEQUENCES ∗ , 2009 .

[26]  A. Teplyaev.  Spectral Analysis on Infinite Sierpiński Gaskets , 1998 .

[27]  R. Grigorchuk.  Degrees of Growth of Finitely Generated Groups, and the Theory of Invariant Means , 1985 .

[28]  Jean-François Quint.  Harmonic analysis on the Pascal graph , 2009 .

[29]  R. I. Grigorchuk.  Branch groups , 2005, math/0510294.

[30]  F. Chung.  Diameters and eigenvalues , 1989 .

[31]  Igor Pak,et al.  Groups of Intermediate Growth, an Introduction , 2008 .

[32]  Noga Alon,et al.  lambda1, Isoperimetric inequalities for graphs, and superconcentrators , 1985, J. Comb. Theory, Ser. B.

[33]  Assaf Naor,et al.  The Euclidean Distortion of the Lamplighter Group , 2007, Discret. Comput. Geom..

[34]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[35]  Alexander Teplyaev,et al.  Laplacians on the basilica Julia set , 2008, 0802.3248.

[36]  Zoran Sunic,et al.  Groups St Andrews 2005: Self-similarity and branching in group theory , 2007 .

[37]  Volodymyr Nekrashevych,et al.  Self-Similar Groups , 2005, 2304.11232.

[38]  Gideon Schechtman,et al.  DIAMOND GRAPHS AND SUPER-REFLEXIVITY , 2009 .

[39]  Seonhee Lim,et al.  Volume entropy of hyperbolic buildings , 2009, 0902.1168.

[40]  Saïd Amghibech.  Bounds for the largest p-Laplacian eigenvalue for graphs , 2006, Discret. Math..

[41]  La Harpe,et al.  Topics in Geometric Group Theory , 2000 .

[42]  T. Nagnibeda,et al.  Schreier graphs of the Basilica group , 2009, 0911.2915.