A generalization of Beck’s Zero-divisor graph: The graph of type 𝒩 associated with a ℤ2-graded ring

Source abstract

Let [Formula: see text] be a commutative [Formula: see text]-graded ring with unity and [Formula: see text] an ideal of [Formula: see text]. Define the [Formula: see text]-function [Formula: see text] by [Formula: see text], where [Formula: see text]. We investigate a new graph associated with [Formula: see text] called the graph of type [Formula: see text] relative to [Formula: see text], denoted by [Formula: see text]. This new graph along with its properties is established using the [Formula: see text]-function which generalizes the notion of the norm of complex numbers. The graph [Formula: see text] represents a significant generalization of Beck’s zero-divisor graph as shown in many parts of the paper. Mainly, our goal is to explore the relationship between different algebraic properties of [Formula: see text] and graphic properties of [Formula: see text] such as connectivity, girth, cliques and dominating sets. Moreover, we perform a similar study concerning the subgraph [Formula: see text] of [Formula: see text] obtained by deleting the zero vertex and all edges incident to [Formula: see text]. Furthermore, we investigate the relationship between the graphs of type [Formula: see text] relative to different ideals [Formula: see text] of [Formula: see text]. One extra important task is a determination of a relation between our new graph and Beck’s zero-divisor graph through different aspects of Graph Theory.