Upper ideal relation graphs associated to rings

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