Simple-intersection graphs of rings

Abstract

Let R be a ring with unity. In this paper, we introduce a new graph associated with R called the simple-intersection graph of R, denoted by GS(R). The vertices of GS(R) are the nonzero ideals of R, and two vertices are adjacent if and only if their intersection is a nonzero simple ideal. We study the interplay between the algebraic properties of R, and the graph properties of GS(R) such as connectedness, bipartiteness, girth, dominating sets, etc. Moreover, we determine the precise values of the girth and diameter of GS(R), as well as give a formula to compute the clique and domination numbers of GS(R).

Keywords

• simple-intersection graph
• rings
• ideals
• clique
• girth
• dominating number
• cycles
• diameter
• connected graph
• complete graph