Related papers: Circulant graphs as an example of discrete quantum…
A finite group $G$ is called $C$-quasirandom (by Gowers) if all non-trivial irreducible complex representations of $G$ have dimension at least $C$. For any unit $\ell^{2}$ function on a finite group we associate the quantum probability…
We prove quantum ergodicity and quantum mixing for sequences of finite Schreier graphs converging to an infinite Cayley graph whose adjacency operator has absolutely continuous spectrum. Under Benjamini-Schramm convergence (or strong…
In this paper, we aim to address the open questions raised in various recent papers regarding characterization of circulant graphs with three or four distinct eigenvalues in their spectra. Our focus is on providing characterizations and…
We investigate which finite Cayley graphs admit a quantum ergodic eigenbasis, proving that this holds for any Cayley graph on a group of size $n$ for which the sum of the dimensions of its irreducible representations is $o(n)$, yet there…
We investigate the equidistribution of the eigenfunctions on quantum graphs in the high-energy limit. Our main result is an estimate of the deviations from equidistribution for large well-connected graphs. We use an exact field-theoretic…
We show that almost all circulant graphs have automorphism groups as small as possible. Of the circulant graphs that do not have automorphism group as small as possible, we give some families of integers such that it is not true that almost…
Consider a sequence of finite regular graphs (GN) converging, in the sense of Benjamini-Schramm, to the infinite regular tree. We study the induced quantum graphs with equilateral edge lengths, Kirchhoff conditions (possibly with a non-zero…
In the past few decades, quantum algorithms have become a popular research area of both mathematicians and engineers. Among them, uniform mixing provides a uniform probability distribution of quantum information over time which attracts a…
Jiang, Tidor, Yao, Zhang, and Zhao recently showed that connected bounded degree graphs have sublinear second eigenvalue multiplicity (always referring to the adjacency matrix). This result was a key step in the solution to the problem of…
The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system, whose hamiltonian is identical to the adjacency matrix of a…
We propose a version of the Quantum Ergodicity theorem on large regular graphs of fixed valency. This is a property of delocalization of "most" eigenfunctions. We consider expander graphs with few short cycles (for instance random large…
We consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are equivalent properties for such Cayley graphs, even if…
An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malni\v{c}, Mart\'{i}nez and…
We consider the family of undirected Cayley graphs associated with odd cyclic groups, and study statistics for the eigenvalues in their spectra. Our results are motivated by analogies between arithmetic geometry and graph theory.
New criteria for which Cayley graphs of cyclic groups of any order can be completely determined--up to isomorphism--by the eigenvalues of their adjacency matrices is presented. Secondly, a new construction for pairs of nonisomorphic Cayley…
We give three different proofs of the main result of Anantharaman-Le Masson, establishing quantum ergodicity -- a form of delocalization --for eigenfunctions of the laplacian on large regular graphs of fixed degree. These three proofs are…
We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakonski et al (J. Phys. A, v. 34, 9303-9317 (2001)). As observables we take the L^2…
Chung, Graham, and Wilson proved that a graph is quasirandom if and only if there is a large gap between its first and second largest eigenvalue. Recently, the authors extended this characterization to k-uniform hypergraphs, but only for…
Let $G$ be a finite group and let $S$ be an inverse-closed subset of $G$ not containing the identity. The Cayley graph $\mathrm{Cay}(G,S)$ has vertex set $G$, where two vertices $x$ and $y$ are adjacent if and only if $x^{-1}y \in S$.…
We consider random $d$-regular graphs on $N$ vertices, with degree $d$ at least $(\log N)^4$. We prove that the Green's function of the adjacency matrix and the Stieltjes transform of its empirical spectral measure are well approximated by…