On Some Graphs Associated to Commutative Semirings

Abstract

Similar to the case of commutative rings, we define the notion of a zero-divisor graph of a commutative semiring R with respect to an ideal I of R, denoted by \({\Gamma_I(R)}\). That is an undirected graph whose vertex set is the set \({\{x \in R {\setminus} I | xy \in I {\rm for some} y \in R {\setminus} I\}}\) with distinct vertices x and y adjacent if and only if xy ∈ I. We discuss when \({\Gamma_I(R)}\) is r-partite. We also give some results on the subgraphs and the parameters of \({\Gamma_I(R)}\). In addition, we apply these results to the annihilating-ideal graph of R with respect to I, denoted by \({\mathbb{AG}_I(R)}\), as an example (special case) of \({\Gamma_I(R)}\). It is also shown that for I a radical ideal of \({R, \Gamma_I(R)}\) is isomorphic to a subgraph of \({\mathbb{AG}_I(R)}\) which naturally asserts some known results (properties) between them interchangeably. Finally, we provide some counterexamples and construct a semiring whose ideal-based zero-divisor graph has a cut-point and more than one bridge, which is in contrast to the ring case.