Some properties of a graph associated with a non-quasi-local weakly factorial domain and a supergraph of it

Let [Formula: see text] be a weakly factorial domain which admits at least two maximal ideals. We use [Formula: see text] to denote the set of all primary elements of [Formula: see text] and denote [Formula: see text] by [Formula: see text]. Let [Formula: see text]. We denote the subset of [Formula: see text] consisting of all [Formula: see text] such that [Formula: see text] does not belong to the Jacobson radical of [Formula: see text] by [Formula: see text]. With [Formula: see text], in this paper, we associate an undirected graph denoted by [Formula: see text] whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In Secs. 2 and 3 of this paper, we discuss results regarding the connectedness, the girth, and the clique number of [Formula: see text] and study the interplay between graph properties of [Formula: see text] and the properties of [Formula: see text]. In Secs. 4 and 5 of this paper, we consider a supergraph of [Formula: see text], denoted by [Formula: see text] whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and study some graph properties of [Formula: see text].