On Weakly S-Prime Ideal Graph Of a Finite Commutative Ring

Let ℋ be a finite commutative ring with unity. Let ℐ be a proper ideal of ℋ and 𝒮 is the multiplicative closed subset of ℋ which is disjoint with ℐ. The weakly S-prime ideal graph denoted by Gℐ(ℋ) is the undirected graph whose vertex set is the set of elements 𝔢 of ℋ such that the non-zero product ef is in ℐ and either se is in ℐ or sf is in ℐ for some f in ℋ and the two distinct vertices 𝔢 and 𝔣 are connected by an edge if and only if either se is in ℐ or sf is in ℐ for some s in 𝒮. The purpose of this article is to investigate the graph theoretic properties of the weakly S-prime ideal graph associated with ℋ. This study focuses on rings of order 2𝔭, 3𝔭 and 𝔭𝔮, where 𝔭 and 𝔮 are distinct primes. For these rings, the weakly S-prime ideal graph is a special type of graph and it is explained with examples. Furthermore, the graph theoretic concept of the weakly S-prime ideal graph Gℐ(ℋ) namely its girth, diameter, radius and size are studied. The relation between the weakly S-prime ideal graph and annihilator ideal graph associated with a ring of order 2𝔭 is described and it is proved that these two graphs are isomorphic.