Integral and distance integral Cayley graphs over generalized dihedral groups

Abstract

A graph is said to be integral (resp. distance integral) if all the eigenvalues of its adjacency matrix (resp. distance matrix) are integers. Let H be a finite abelian group, and let \({\mathscr {H}}=\langle H,b\,|\,b^2=1,bhb=h^{-1},h\in H\rangle \) be the generalized dihedral group of H. The contribution of this paper is threefold. Firstly, based on the representation theory of finite groups, we obtain a necessary and sufficient condition for a Cayley graph over \({\mathscr {H}}\) to be integral, which naturally contains the main results obtained in Lu et al. (J Algebr Comb 47:585–601, 2018). Secondly, a closed-form decomposition formula for the distance matrix of Cayley graphs over any finite groups is derived. As applications, a necessary and sufficient condition for the distance integrality of Cayley graphs over \({\mathscr {H}}\) is displayed. Some simple sufficient (or necessary) conditions for the integrality and distance integrality of Cayley graph are exhibited, respectively, from which several infinite families of integral and distance integral Cayley graphs over \({\mathscr {H}}\) are constructed. And lastly, some necessary and sufficient conditions for the equivalence of integrity and distance integrity of Cayley graphs over generalized dihedral groups are obtained.