On a Generalized Zero-divisor Graph of a Commutative Ring with Respect to an ideal - Baruah, Priyanka Pratim; Patra, Kuntala (Communications in Mathematics and Applications, 2017)
In this paper, we generalize the notion of the ideal-based zero-divisor graph of a commutative ring. Let \(R\) be a commutative ring and let \(I\) be an ideal of \(R\). Here, we define a generalized zero-divisor graph of \(R\) with respect to \(I\) and denote this graph by \(Γ_I^G(R)\). We show that \(Γ_I^G(R)\) is connected with diameter at most three. If \(Γ_I^G(R)\) has a cycle, we show that the girth of \(Γ_I^G(R)\) is at most four. Also, we investigate the existence of cut vertices of \(Γ_I^G(R)\). Moreover,we examine certain situations when \(Γ_I^G(R)\) is a complete bipartite graph.