Line graphs associated to Von Neumann regular graphs of rings

Abstract

Let [Formula: see text] be a commutative ring with unity. Taloukolaei and Sahebi [Von Neumann regular graphs associated with rings, Discrete Math. Algorithms Appl. 10(3) (2018) 1850029, doi:10.1142/S1793830918500295] introduced the Von Neumann regular graph [Formula: see text] of a ring [Formula: see text] whose vertices are the elements of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is a Von Neumann regular element of [Formula: see text]. In this paper, we investigate and determine some graph theoretic properties of the line graph [Formula: see text] associated to [Formula: see text]. We give some characterization results regarding the completeness, bipartiteness, traversability, diameter and girth. We also prove Beck’s conjecture for [Formula: see text]. Finally we characterize rings having the planarity, the outerplanarity and also being the ring graph of the line graphs associated to Von Neumann regular graphs [Formula: see text] of rings.

Graphs Defined on Rings: A Review

“…The line graph of a graph is a well-studied derived graph of a graph, and, as already known, several properties of the line graph of a graph are interrelated with the properties of the graph. In this regard, the line graphs of the unit graphs associated with the finite commutative rings were exclusively studied in [259][260][261]. The basic properties of the line graph of the unit graph of finite commutative rings such as the diameter, girth, clique, and chromatic number, along with some classifications of rings whose unit graphs are planar, as well as the Hamiltonian, were given in [260].…”

“…An extended investigation on the line graph of the unit graph associated with finite commutative rings was performed in [259], where characterisations of the line graphs of the unit graphs of rings on the basis of their structural properties such as the completeness, bipartiteness, traversability, diameter, girth, and chromatic number were obtained. Additionally, the domination number of this line graph of the unit graph of rings was computed in [234], along with the domination number of the unit graphs of rings.…”