On Another Class of Strongly Perfect Graphs

Abstract

For a commutative ring R with unity, the associate ring graph, denoted by AG(R), is a simple graph with vertices as nonzero elements of R and two distinct vertices are adjacent if they are associates. The graph AG(R) contains components equal in number to the number of distinct orbits, except for the orbit of an element 0. Moreover, each component is a complete graph. An important finding is that this is a class of strongly perfect graphs. In this article we describe the structure of the associate ring graph of the ring of integers modulo n, denoted by AG(Zn). We carried out computer experiments and provide a program for the same. We further characterize cases in which AG(Zn), its complement AG(Zn)¯, and their line graphs are planar, ring graphs, and outerplanar. We also discuss the properties of the associate ring graph of a commutative ring R with unity.

Algorithm for Constructing Total Graph of Commutative Ring

“…Python was created by Guido van Rossum in the late 1980s and early 1990s at the National Research Institute for Mathematics and Computer Science in Netherlands (Saabith et al, 2019). In the graphic concept of ring, several researchers have used python algorithms such as in (Tapanyo et al, 2022), (Kansal et al, 2022), (Hartati & Kurniawan, 2023), and (Tian & Li, 2022). In previous research, no algorithm was given to construct the total graph.…”