Unity Product Graph of Some Commutative Rings

TL;DR

This paper introduces a new unity product graph for commutative rings, analyzing its properties and relationships with the ring's structure, including connectivity, isolated vertices, and graph invariants.

Contribution

It defines the unity product graph for commutative rings and explores its fundamental properties, including connectivity, trivial cases, and various graph invariants, linking algebraic and graph-theoretic concepts.

Findings

Unity product graph is disconnected if it has at least 2 vertices.

Complement of the unity product graph is connected.

Some rings have trivial or complete unity product graphs.

Abstract

A graph is an instrument which is extensively utilized to model various problems in different fields. Up to date, many graphs have been developed to represent algebraic structures, particularly rings in order to study their properties. In this article, by focusing on commutative ring $ R $, we introduce a new notion of unity product graph associated with $ R $ and its complement. In addition, we prove that if the number of vertices of the unity product graph is at least 2, then the graph is disconnected, while its complement graph is connected. Furthermore, it is shown that there are some commutative rings with such as Boolean ring and the Cartesian product of Boolean rings in which their associated unity product graphs are trivial. Consequently, some results are established to determine the number of isolated vertices in unity product graph. We also characterize commutative rings with unity in which their associated unity product and complement unity product graphs are empty graph and complete graph, respectively. Finally, we prove some results on the properties of the unity product graph and its complement in terms of girth, diameter, radius, dominating number, chromatic number and clique number as well as planarity and Hamiltonian.