Lamplighter groups, de Bruijn graphs, spider-web graphs and their spectra

TL;DR

This paper studies the spectral properties of spider-web graphs, showing they are Schreier graphs of lamplighter groups, and identifies their infinite limits with Cayley graphs of these groups, connecting graph theory and group theory.

Contribution

It establishes that spider-web graphs are Schreier graphs of lamplighter groups and computes their spectra, linking finite graph structures to infinite group limits.

Findings

Spider-web graphs are Schreier graphs of lamplighter groups.

Spectra of these graphs are explicitly computed.

Infinite limits of the graphs correspond to Cayley graphs of lamplighter groups.

Abstract

We describe the infinite family of spider-web graphs $S_{k,M,N }$, $k \geq 2$, $M \geq 1$ and $N \geq 0$, studied in physical literature as tensor products of well-known de Brujin graphs $B_{k,N}$ and cyclic graphs $C_M$ and show that these graphs are Schreier graphs of the lamplighter groups $L_k = Z/kZ \wr Z$. This allows us to compute their spectra and to identify the infinite limit of $S_{k,M,N}$, as $N, M \to\infty$, with the Cayley graph of the lamplighter group $L_k$. This is the final version of the article, taking in account comments from the referees and with an extended introduction.