On Ideal Based Weakly Zero Divisor Graph and its Properties

Abstract

For a commutative ring S having J as an ideal, the weakly zero divisor graph of S, denoted by WΓ(S), has the zero divisors of S as its vertices, and two distinct vertices a and b are adjacent if there exist r, s ∈ S \ {0} such that ar, bs and rs are all zero. An ideal based weakly zero divisor graph of S, denoted by WΓ_J (S), is the graph with the vertex set {a ∈ S \ J : ab ∈ J for some b ∈ S \ J} and the edge set {(a, b) : ar ∈ J, bs ∈ J and rs ∈ J for some r, s ∈ S \ J}. The graph WΓ_J (S) contains the ideal based zero divisor graph Γ_J (S) as a subgraph and is identical to WΓ(S) if J = {0}. In this article, the clique number and the connectivity of the graph WΓ_J (S) are determined and the bounds on these numbers are provided. The clique number and connectivity of the graph WΓ_J (S) are related to the clique number and connectivity of the graph WΓ(S / J).