On Cayley sum graph of ideals of a ring

Let [Formula: see text] be a commutative ring with unity, [Formula: see text] be the set of all ideals of [Formula: see text] and [Formula: see text] be a subset of [Formula: see text]. Cayley sum graph of ideals of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent ([Formula: see text]), whenever [Formula: see text] or [Formula: see text], for some ideal [Formula: see text] in [Formula: see text]. In this paper, we determine the girth, chromatic number, clique number, perfect code, and perfectness of the Cayley sum graphs [Formula: see text] and [Formula: see text] (where [Formula: see text] denotes the set of all nonzero maximal ideals of [Formula: see text]). Finally, we classify the family of rings whose Cayley sum graphs are split, co-graph, threshold, chordal and claw-free.