A graph associated with tri-potent elements of commutative ring R

Abstract

In this paper, we introduce the tri-potent graph of a commutative ring R, denoted by TP(R), where two distinct vertices x and y in R are adjacent if and only if (x + y)³ = x + y. We conduct a comprehensive investigation of the graphical structural properties of tri-potent graph of a commutative ring R, including its diameter, connectedness, and size. It is shown that the tri-potent graph of a commutative ring R contains cycles with girth 3 and has no end vertices. Furthermore, we describe a significant spanning subgraph of the tri-potent graph of a commutative ring R and analyze the degree of each vertex in detail. Finally, we establish that, for a specific local ring, tri-potent graph of a commutative ring R forms a 3-partite graph, discuss its planarity, and determine its independence domination number.