On Graph Embeddings of the Weak Zero-Divisor Graphs of Commutative Rings

Let [Formula: see text] be a commutative ring, and let [Formula: see text] denote the set of all nonzero elements of [Formula: see text]. The weak zero-divisor graph of [Formula: see text], denoted by [Formula: see text], is an undirected graph whose vertex set consists of the elements in [Formula: see text]. Two distinct vertices [Formula: see text] and [Formula: see text] are connected by an edge if and only if there exists a positive integer [Formula: see text] such that [Formula: see text]. In this study, we investigate finite commutative rings [Formula: see text] whose associated weak zero-divisor graphs [Formula: see text] belong to certain well-established classes of graphs. Specifically, we provide a classification of finite rings [Formula: see text] for which the graph [Formula: see text] is a unicyclic graph, a tree, a split graph, a planar graph, or an outerplanar graph. In addition, we explore the conditions under which [Formula: see text] is a toroidal graph.