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Generic spectrum of the weighted Laplacian operator on Cayley graphs

Cristian F. Coletti, Lucas R. de Lima, Diego S. de Oliveira, Marcus A. M. Marrocos

Abstract

In this paper, we investigate the spectrum of a class of weighted Laplacians on Cayley graphs and determine under what conditions the corresponding eigenspaces are generically irreducible. Specifically, we analyze the spectrum on left-invariant Cayley graphs endowed with an invariant metric, and we give some criteria for generically irreducible eigenspaces. Additionally, we introduce an operator that is comparable to the Laplacian and show that the same criterion holds.

Generic spectrum of the weighted Laplacian operator on Cayley graphs

Abstract

In this paper, we investigate the spectrum of a class of weighted Laplacians on Cayley graphs and determine under what conditions the corresponding eigenspaces are generically irreducible. Specifically, we analyze the spectrum on left-invariant Cayley graphs endowed with an invariant metric, and we give some criteria for generically irreducible eigenspaces. Additionally, we introduce an operator that is comparable to the Laplacian and show that the same criterion holds.

Paper Structure

This paper contains 8 sections, 9 theorems, 63 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Criterion for G-simplicity on Cayley graphs
  4. Proof of the main criterion for finite Cayley graphs
  5. Representations of degree 1
  6. Representation of degree 2
  7. Representations of degree 3
  8. The operator L w

Key Result

Theorem \oldthetheorem

A finite Cayley graph $\mathcal{C}(G,S)$ admits a weight $w \in \mathcal{L}_S$ such that $\Delta_w$ is real $G$-simple if, and only if, the following conditions are simultaneously satisfied Moreover, the existence implies that $\mathcal{W}_S$, the set of all invariant weights $w \in \mathcal{L}_S$ that turns $\Delta_w$ into a real $G$-simple operator, is a residual set in $\mathcal{L}_S$.

Figures (2)

  • Figure 1: Tetrahedrons associated with the Cayley graphs of the Klein four-group generated by the set $S=\{a,b,ab\}$ (left) and with the cyclic group $\mathcal{C}(\mathbf{C_4}, \{r,r^2,r^3\})$ (right) (see Examples \ref{['ex:cyclic.dir']} and \ref{['ex:Klein-four']}).
  • Figure 2: The truncated cube $\mathcal{C}(\mathbf{S}_4,S)$ (left) and $\mathcal{C}(\mathbf{S}_4,T)$ with $T= \{\tau,\sigma^{\pm1},\eta^{\pm1}\}$ (right), the graph union of the truncated cube with the rhombicuboctahedron.

Theorems & Definitions (29)

  • Theorem \oldthetheorem
  • Definition : Cayley graph
  • Lemma \oldthetheorem
  • proof
  • Remark 1
  • Definition
  • Proposition \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • proof
  • ...and 19 more