Which Cayley Graphs are Integral?

Abstract

Let $G$ be a non-trivial group, $S\subseteq G\setminus \{1\}$ and $S=S^{-1}:=\{s^{-1} \;|\; s\in S\}$. The Cayley graph of $G$ denoted by $\Gamma(S:G)$ is a graph with vertex set $G$ and two vertices $a$ and $b$ are adjacent if $ab^{-1}\in S$. A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine all connected cubic integral Cayley graphs. We also introduce some infinite families of connected integral Cayley graphs.

Integral Cayley graphs over dihedral groups

“…Let ρ : G → GL(V) be a representation. The character χ ρ : G → C of ρ is defined by setting χ ρ (g) = T r(ρ(g)) for g ∈ G, where T r(ρ(g)) is the trace of the representation matrix of ρ(g) with respect to some basis of V. The degree of the character χ ρ is just the degree of ρ, which equals to χ ρ (1). A subspace W of V is said to be G-invariant if ρ(g)w ∈ W for each g ∈ G and w ∈ W. If W is a G-invariant subspace of V, then the restriction of ρ on W, i.e., ρ |W : G → GL(W), is a representation of G on W. Obviously, {1} and V are always G-invariant subspaces, which are called trivial.…”

“…In 2005, So [15] gave a complete characterization of integral circulant graphs. Later, Abdollahi and Vatandoost [1] showed that there are exactly seven connected cubic integral Cayley graphs in 2009. About the same year, Klotz and Sander [11] proved that, for an abelian group G, if the Cayley graph X(G, S ) is integral then S belongs to the Boolean algebra B(F G ) generalized by the subgroups of G. Moreover, they conjectured that the converse is also true, which has been proved by Alperin and Peterson [2].In 2014, Cheng, Terry and Wong (cf.[6], Corollary 1.2) presented that the normal Cayley graphs over symmetric groups are integral (a Cayley graph is said to be normal if its generating set S is closed under conjugation).…”

On the eigenvalues of certain Cayley graphs and arrangement graphs

“…, n} and S ⊆ S n be closed under conjugation. Since In general, if S is not closed under conjugation, then the eigenvalues of Γ(S n , S) may not be integers [13] (see also [1,17,20] for related results on the eigenvalues of certain Cayley graphs). Problem 1.3.…”

“…, (1 n)}. It was conjectured by Abdollahi and Vatandoost [1] that the eigenvalues of Γ(S n , Cy(2)) are integers, and contains all integers in the range from −(n − 1) to n − 1 (with the sole exception that when n = 2 or 3, zero is not an eigenvalue of Γ(S n , Cy(2)). The second part of the conjecture was proved by Krakovski and Mohar [17].…”

Spectrum and genus of commuting graphs of some classes of finite rings

“…The notion of integral graph was introduced by Harary and Schwenk [11] in the year 1974. Since then many mathematicians have considered integral graphs, see for example [2,12,18]. In [9,15], the authors have determined several groups whose commuting graphs are integral.…”