Study of Prime Graph of a Ring

Kishor Pawar, Sandeep Joshi

Abstract

The notion of a prime graph of a ring $R$, ($PG(R)$) was first introduced by {\sc S. Bhavanari and his coauthors} in [1]. In this paper, we introduce the notion of `Complement of a Prime Graph of a Ring $R$', denote it by $(PG(R))^c$ and find the degree of vertices in $PG(R)$ and $(PG(R))^c$ for the ring $\mz_n$ and the number of triangles in $PG(R)$ and $(PG(R))^c$. It is proved that for any $n \geq 6$ which not a prime then $gr(PG(\mz_n ))=3$. If $n$ is any prime number or $n=4$ then $gr(PG(\mz_n))= \infty$.