CAYLEY SUM GRAPHS OF IDEALS OF A COMMUTATIVE RING

Let R be a commutative ring, I(R) be the set of all ideals of R and S be a subset of I*(R) = I(R) \ {0}. We define a Cayley sum digraph of ideals of R, denoted by (Cay) over right arrow (+) (I(R), S), as a directed graph whose vertex set is the set I(R) and, for every two distinct vertices I and J, there is an arc from I to J, denoted by I -> J, whenever I + K = J, for some ideal K in S. Also, the Cayley sum graph Cay(+) (I(R), S) is an undirected graph whose vertex set is the set I(R) and two distinct vertices I and J are adjacent whenever I + K = J or J + K = I, for some ideal K in S. In this paper, we study some basic properties of the graphs (Cay) over right arrow (+) (I(R), S) and Cay(+) (I(R), S) such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of Cay(+) (I(R), S) and also we provide some characterization for rings R whose Cayley sum graphs have genus one.