A study of upper ideal relation graphs of rings

Abstract

Let R be a ring with unity. The upper ideal relation graph of the ring R is the 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 such that the ideals (x) and (y) contained in the ideal (z). In this article, we obtain the girth, minimum degree and the independence number of . We obtain a necessary and sufficient condition on R, in terms of the cardinality of their principal ideals, such that the graph is planar and outerplanar, respectively. For a non-local commutative ring , where Ri is a local ring with maximal ideal and , we prove that the graph is perfect if and only if and each is a principal ideal. We also discuss all the finite rings R such that the graph is Eulerian. Moreover, we obtain the metric dimension and strong metric dimension of , when R is a reduced ring. Finally, we determine the vertex connectivity, automorphism group, Laplacian and the normalized Laplacian spectrum of . We classify all the values of n for which the graph is Hamiltonian.