Enumeration of cubic Cayley graphs on dihedral groups

Abstract

Let $p$ be an odd prime, and $D_{2p}=\langle a,b\mid a^p=b^2=1,bab=a^{-1}\rangle$ the dihedral group of order $2p$. In this paper, we completely classify the cubic Cayley graphs on $D_{2p}$ up to isomorphism by means of spectral method. By the way, we show that two cubic Cayley graphs on $D_{2p}$ are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley graphs on $D_{2p}$ by using Gauss' celebrated law of quadratic reciprocity.Comment: 15 pages, 0 figur

Eigenvalues of Cayley graphs

“…The following two theorems were proved by Abdollahi et al in [3]. Huang et al [154] also posed the question of classifying and enumerating connected cubic Cayley graphs on general dihedral groups and determining which of them have the Cay-DS property.…”

“…is the dihedral group of order 2n ≥ 4 and S is a subset of D 2n \ {1} with S −1 = S. The Cayley graph Cay(D 2n , S) is called a dihedrant in the literature, and its eigenvalues can be computed using Theorem 2.2 and the character table of D 2n (see[184, Section 7.5] and[154, Corollary 2.7]). Recently, Gao and Luo[123] gave a simpler way to compute the spectra of Cay(D 2n , S) using a more general result (see Theorem 11.1) on eigenvalues of semi-Cayley (bi-Cayley) graphs on abelian groups.…”