Integral Cayley Graphs over Dihedral Groups

TL;DR

This paper characterizes when Cayley graphs over dihedral groups are integral, providing necessary and sufficient conditions, and identifies all integral Cayley graphs over dihedral groups of prime order.

Contribution

It offers a complete characterization of integral Cayley graphs over dihedral groups, including prime order cases, and introduces simple conditions based on Boolean algebra.

Findings

Necessary and sufficient condition for integrality over D_n

Infinite classes of integral Cayley graphs over D_n

Complete classification for D_p with prime p

Abstract

In this paper, we give a necessary and sufficient condition for the integrality of Cayley graphs over the dihedral group $D_n=\langle a,b\mid a^n=b^2=1,bab=a^{-1}\rangle$. Moreover, we also obtain some simple sufficient conditions for the integrality of Cayley graphs over $D_n$ in terms of the Boolean algebra of $\langle a\rangle$, from which we find infinite classes of integral Cayley graphs over $D_n$. In particular, we completely determine all integral Cayley graphs over the dihedral group $D_p$ for a prime $p$.