On Spectra and Spectral Measures of Schreier and Cayley Graphs

2019

We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of Lebesgue measure zero for associated Jacobi operators if the subshift is aperiodic. Our class covers all simple Toeplitz subshifts as well as all Sturmian subshifts. We apply our results to the spectral theory of Schreier graphs for uncountable families of groups acting on rooted trees.

Transactions of the American Mathematical Society

We show that the lamplighter group L = Z/2Z Z has a system of generators for which the spectrum of the discrete Laplacian on the Cayley graph is a union of an interval and a countable set of isolated points accumulating to a point outside this interval. This is the first example of a group with infinitely many gaps in the Spectrum of its Cayley graph. The result is obtained by a careful study of spectral properties of a one-parametric family a + a -1 + b + b -1 -µc of convolution operators on L where µ is a real parameter. Our results show that the spectrum is a pure point spectrum for each value of µ, the eigenvalues are solutions of algebraic equations involving Chebyshev polynomials of the second kind, and the topological structure of the spectrum makes a bifurcation when the parameter µ passes the points 1 and -1. Namely if |µ| ≤ 1 the spectrum is an interval while when |µ| > 1 it is a union of an interval and a countable set of points accumulating to a point outside the interval.

Groups, Graphs and Random Walks

There is a recently discovered connection between the spectral theory of Schrödinger operators whose potentials exhibit aperiodic order and that of Laplacians associated with actions of groups on regular rooted trees, as Grigorchuk's group of intermediate growth. We give an overview of corresponding results, such as different spectral types in the isotropic and anisotropic cases, including Cantor spectrum of Lebesgue measure zero and absence of eigenvalues. Moreover, we discuss the relevant background as well as the combinatorial and dynamical tools that allow one to establish the afore-mentioned connection. The main such tool is the subshift associated to a substitution over a finite alphabet that defines the group algebraically via a recursive presentation by generators and relators. Contents Introduction 1. Subshifts and aperiodic order in one dimension 2. Schrödinger operators with aperiodic order 2.1. Constancy of the spectrum and the integrated density of states (IDS) 2.2. The spectrum as a set and the absolute continuity of spectral measures 2.3. Aperiodic order and discrete random Schrödinger operators 3. The substitution τ , its finite words Sub τ and its subshift (Ω τ , T) 3.1. The substitution τ and its subshift: basic features 3.2. The main ingredient for our further analysis: n-partition and n-decomposition 3.3. The maximal equicontinuous factor of the dynamical system (Ω τ , T) 3.4. Powers and the index (critical exponent) of Sub τ 3.5. The word complexity of Sub τ 3.6. Generating the fixed point η by an automaton 3.7. Replacing τ by a primitive substitution 4. Background on graphs and dynamical systems 5. Grigorchuk's group G, its Schreier graphs and the associated Laplacians 5.1. Grigorchuk's group G 5.2. The Schreier graphs of G and the dynamical system (X, G) 5.3. Laplacians associated to the Schreier graphs of G 6. The connection between (X, G) and (Ω τ , T) 7. Spectral theory of the M ξ for ξ ∈ ∂T 8. Integrated density of states and Kesten-von-Neumann-Serre spectral measure References

Comptes Rendus Mathematique, 2006

European Journal of Combinatorics, 2012

For every infinite sequence ω = x 1 x 2 . . ., with x i ∈ {0, 1}, we construct an infinite 4-regular graph X ω . These graphs are precisely the Schreier graphs of the action of a certain self-similar group on the space {0, 1} ∞ . We solve the isomorphism and local isomorphism problems for these graphs, and determine their automorphism groups. Finally, we prove that all graphs X ω have intermediate growth.

Geometriae Dedicata, 2006

We calculate the spectra and spectral measures associated to random walks on restricted wreath products G wr Z, with G a finite group by calculating the Kesten-von Neumann-Serre spectral measures for the random walks on Schreier graphs of certain groups generated by automata. This generalises the work of Grigorchuk andŻuk on the lamplighter group. In the process we characterise when the usual spectral measure for a group generated by automata coincides with the Kestenvon Neumann-Serre spectral measure.

2021

For a simple graph G, the generalized adjacency matrix Aα(G) is defined as Aα(G) = αD(G) + (1− α)A(G), α ∈ [0, 1], where A(G) is the adjacency matrix and D(G) is the diagonal matrix of vertex degrees of G. This matrix generalises the spectral theories of the adjacency matrix and the signless Laplacian matrix of G. In this paper, we find Aα-spectrum of the joined union of graphs in terms of spectrum of adjacency matrices of its components and the zeros of the characteristic polynomials of an auxiliary matrix determined by the joined union. We determine the Aα-spectrum of join of two regular graphs, the join of a regular graph with the union of two regular graphs of distinct degrees. As an applications, we investigate the Aα-spectrum of certain power graphs of finite groups.

arXiv (Cornell University), 2024

The present paper investigates the limit G-space J G generated by the self-similar action of automatic groups on a regular rooted tree. The limit space J G is the Gromov-Hausdorff limit of the family of Schreier graphs Γn; therefore, J G can be approximated by Schreier graphs on level n-th when n tends to infinity. We propose a computer program whose code is written in Wolfram language computes the adjacency matrix of Schreier graph Γn at each specified level of the regular rooted tree. In this paper, the Schreier graphs corresponding to each automatic group is computed by applying the program to some collection of automata groups, including classic automatic groups.

arXiv: Combinatorics, 2018

The power graph $\mathcal{G}(G)$ of a group $G$ is a simple graph whose vertices are the elements of $G$ and two distinct vertices are adjacent if one is a power of other. In this paper, we investigate the Laplacian spectrum of the power graph $\mathcal{G}(\mathbb{Z}_{p^m}^n)$ of finite abelian $p$-group $\mathbb{Z}_{p^m}^n$. In particular, we prove that the spectrum of group $\mathbb{Z}_{p^m}^n$ is contained in the Laplacian spectrum of graph $\mathcal{G}(\mathbb{Z}_{p^m}^n)$. For a finite abelian group $G$ whose power graph $\mathcal{G}(G)$ is planar, we also prove that the spectrum of group $G$ is contained in the Laplacian spectrum of graph $\mathcal{G}(G)$.

Linear Algebra and its Applications, 1982

The spectrum of a locally finite countable graph is defined. Some theorems known from the theory of spectra of finite graphs are extended to infinite graphs.