Intersection Graph of Graded ideals

TL;DR

This paper introduces and analyzes the intersection graph of graded ideals in graded rings, exploring various graph properties and special cases like faithful, strong, and ordered group gradings.

Contribution

It defines the intersection graph of graded ideals and studies its properties across different types of gradings, extending previous concepts to new algebraic structures.

Findings

Graph connectivity and regularity conditions identified.

Complete and domination properties characterized.

Intersection graphs for specific gradings analyzed.

Abstract

In this article we introduce and study the intersection graph of graded ideals of graded rings. The intersection graph of $G-$graded ideals of a graded ring $(R,G)$ is a simple graph, denoted by $Gr_G(R)$, whose vertices are the nontrivial graded ideals and two ideals are adjacent if they are not trivially intersected. We study graphical properties for these graphs such as connectivity, regularity, completeness, domination numbers, and girth. These intersection graphs for faithful, strong, and first strong gradings are also discussed. In addition, we investigate intersection graphs of $\mathbb{Z}_2-$graded idealization, and we deal with intersection graph of graded ideals when the grading group is an ordered groups.