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A formula for eigenvalues of integral Cayley graphs over abelian groups

Priya, Monu Kadyan

Abstract

Let $Z$ be an abelian group, $ x \in Z$, and $[x] = \{ y : \langle x \rangle = \langle y \rangle \}$. A graph is called integral if all its eigenvalues are integers. It is known that a Cayley graph is integral if and only if its connection set can be express as union of the sets $[x] $. In this paper, we determine an algebraic formula for eigenvalues of the integral Cayley graph when the connection set is $ [x]$. This formula involves an analogue of M$\ddot{\text{o}}$bius function.

A formula for eigenvalues of integral Cayley graphs over abelian groups

Abstract

Let be an abelian group, , and . A graph is called integral if all its eigenvalues are integers. It is known that a Cayley graph is integral if and only if its connection set can be express as union of the sets . In this paper, we determine an algebraic formula for eigenvalues of the integral Cayley graph when the connection set is . This formula involves an analogue of Mbius function.

Paper Structure

This paper contains 3 sections, 9 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Main Theorem

Key Result

Theorem 1.1

(bridges1982rational) Let $Z$ be an abelian group and $S$ be a close under inverse subset of $Z$. The Cayley graph $\text{Cay}(Z, S)$ is integral if and only if $S=[x_1]\cup [x_2]\cup \ldots \cup [x_r]$ for some $x_1, x_2, \ldots , x_r \in S$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2: babai1979spectra
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • ...and 6 more