Let $S$ be a subset of a finite abelian group $G$ . The Cayley sum graph of $G$ with respect to $S$ is a graph whose vertex set is $G$ and two vertices $g$ and $h$ are joined by an edge if and only if $g+h \\in S$ . In this thesis, first we prove some elementary properties of Cayley sum graphs . Also , using the concept of Cayley sum color graphs and representation theory methods , we find the spectrum of real anti-circulant matrices . Finally, we classify simple connected cubic integral Cayley sum graphs .
