Prime Graph of Cartesian Product of Rings

Abstract

Let be a commutative ring. The prime graph of the ring is defined as a graph whose vertex set consists of all elements of and any two distinct vertices x and y are adjacent if and only if or . This graph is denoted by . In this paper we investigate some relations between the chromatic number of prime graph of finite product of commutative rings and the chromatic number of prime graph of these rings. We also obtain some results on the chromatic number of prime graph of the ring .

Algorithm for Constructing Total Graph of Commutative Ring

“…In (2021), Any and Hidayah have researched the girth of the total graph of ℤ 𝑛, denoted by 𝑔𝑟(𝑇 𝛤 (ℤ 𝑛 )). Kalita et al (2014) introduce the prime graph of the commutative Ring ℤ 𝑛, . Prime graph is a graph associated with a ring denoted 𝑃𝐺(𝑅).…”