Integral Cayley Graphs Defined by Greatest Common Divisors

Abstract

An undirected graph is called integral, if all of its eigenvalues are integers. Let $\Gamma =Z_{m_1}\otimes \ldots \otimes Z_{m_r}$ be an abelian group represented as the direct product of cyclic groups $Z_{m_i}$ of order $m_i$ such that all greatest common divisors $\gcd(m_i,m_j)\leq 2$ for $i\neq j$. We prove that a Cayley graph $Cay(\Gamma,S)$ over $\Gamma$ is integral, if and only if $S\subseteq \Gamma$ belongs to the the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. It is also shown that every $S\in B(\Gamma)$ can be characterized by greatest common divisors.

Distance Powers and Distance Matrices of Integral Cayley Graphs over Abelian Groups

“…As in the proof of Theorem 1 it suffices to show that any sum of elementary gcd-sets is a gcd-set. If Γ is cyclic, then B gcd (Γ) = B(Γ) (see Theorem 3 in [13]) and the result follows from Lemma 1. Now let Γ = Z m 1 ⊕ .…”

“…By construction, the elementary gcd-sets are the atoms of the Boolean algebra B gcd (Γ) consisting of all gcd-sets of Γ. According to Theorem 1 in [13], B gcd (Γ) is a Boolean sub-algebra of B(Γ). Hence by Theorem 2, all gcdgraphs Cay(Γ, S), S ∈ B gcd (Γ), are integral.…”

“…This extends an analogous result of Ilić [11] for integral circulant graphs, which are the integral Cayley graphs over cyclic groups. Finally, we show that the class of gcd-graphs, another subclass of integral Cayley graphs over abelian groups (see [13]), is also closed under distance power operations.…”

Integral Sets and Cayley Graphs of Finite Groups

“…The integral Cayley graphs for a cyclic group were determined in [3]. This result was rediscovered [8]; recently, there is renewed interest in the case of abelian groups [6], [7].…”

Groups all of whose undirected Cayley graphs are integral

“…Cayley graphs which are integral were studied by many people (e.g. [1,2,4,7,10,11]). Following [10] we call a group G Cayley integral whenever all undirected Cayley graphs over G are integral.…”