Upper ideal relation graphs associated to rings

Abstract

Let $R$ be a ring with unity. The upper ideal relation graph ${\Gamma }_U(R)$ of the ring $R$ is a simple undirected graph whose vertex set is the set of all non-unit elements of $R$ and two distinct vertices $x,y$ are adjacent if and only if there exists a non-unit element $z\in R$ such that the ideals $(x)$ and $(y)$ contained in the ideal $(z)$. In this article, we classify all the non-local finite commutative rings whose upper ideal relation graphs are split graphs, threshold graphs and cographs, respectively. In order to study topological properties of ${\Gamma }_U(R)$, we determine all the non-local finite commutative rings $R$ whose upper ideal relation graph has genus at most $2$. Further, we precisely characterize all the non-local finite commutative rings for which the crosscap of ${\Gamma }_U(R)$ is either $1$ or $2$.