Cayley sum color and anti-circulant graphs

Let G be a finite group and α:G→R be a real-valued function on G. The Cayley sum color graph Cay+(G,α) is a complete directed graph with vertex set G where each arc (x,y)∈G×G is associated with a color α(xy). If α is the characteristic function on a subset S of G, then the Cayley sum graph Cay+(G,S) is obtained. The anti-circulant matrix associated to a vector v is an n×n matrix whose rows are given by iterations of the anti-shift operator acting on v. We note that a graph is a Cayley sum graph of a cyclic group if and only if it is an anti-circulant graph, a graph whose adjacency matrix is anti-circulant. In this paper, we obtain some results on the isomorphisms, connectivity and vertex transitivity about anti-circulant graphs. We find the spectrum of Cayley sum color graphs of abelian groups and as a result we compute the spectrum of real anti-circulant matrices.