A study of upper ideal relation graphs of rings

Abstract

Let R be a ring with unity. The upper ideal relation graph ${\Gamma }_U(R)$ 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 $z\in R$ 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 ${\Gamma }_U(R)$. We obtain a necessary and sufficient condition on R, in terms of the cardinality of their principal ideals, such that the graph ${\Gamma }_U(R)$ is planar and outerplanar, respectively. For a non-local commutative ring $R\cong R_1\times R_2\times \cdots \times R_n$, where R_i is a local ring with maximal ideal ${\mathcal{M}}_i$ and $n\ge 3$, we prove that the graph ${\Gamma }_U(R)$ is perfect if and only if $n\in \{3,4\}$ and each ${\mathcal{M}}_i$ is a principal ideal. We also discuss all the finite rings R such that the graph ${\Gamma }_U(R)$ is Eulerian. Moreover, we obtain the metric dimension and strong metric dimension of ${\Gamma }_U(R)$, when R is a reduced ring. Finally, we determine the vertex connectivity, automorphism group, Laplacian and the normalized Laplacian spectrum of ${\Gamma }_U({\mathbb{Z}}_n)$. We classify all the values of n for which the graph ${\Gamma }_U({\mathbb{Z}}_n)$ is Hamiltonian.

Keywords:

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

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 05C25
• 05C50

Acknowledgments

The authors are deeply grateful to the referees for the careful reading of the manuscript and helpful suggestions.

Additional information

Funding

The first author gratefully acknowledges for providing financial support to CSIR (09/719(0093)/2019-EMR-I) government of India. The second author sincerely acknowledges for providing financial support to Birla Institute of Technology and Science (BITS) Pilani, India.