Upper ideal relation graphs associated to rings

Let R be a ring with unity. The upper ideal relation graph ΓU(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 ∈ 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 ΓU(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 ΓU(R) is either $1$ or $2$.