A study of upper ideal relation graphs of rings

Abstract

AbstractLet R be a ring with unity. The upper ideal relation graph [Formula: see text] 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 [Formula: see text] 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 [Formula: see text]. We obtain a necessary and sufficient condition on R, in terms of the cardinality of their principal ideals, such that the graph [Formula: see text] is planar and outerplanar, respectively. For a non-local commutative ring [Formula: see text], where Ri is a local ring with maximal ideal [Formula: see text] and [Formula: see text], we prove that the graph [Formula: see text] is perfect if and only if [Formula: see text] and each [Formula: see text] is a principal ideal. We also discuss all the finite rings R such that the graph [Formula: see text] is Eulerian. Moreover, we obtain the metric dimension and strong metric dimension of [Formula: see text], when R is a reduced ring. Finally, we determine the vertex connectivity, automorphism group, Laplacian and the normalized Laplacian spectrum of [Formula: see text]. We classify all the values of n for which the graph [Formula: see text] is Hamiltonian.

Keywords

• Upper ideal relation graph
• principal ideals
• maximal ideals
• metric dimension
• strong metric dimension
• 05C25