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.

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Let S be a commutative semiring with unity. In this paper, we introduce the weakly nilpotent graph of a commutative semiring. The weakly nilpotent graph of S, denoted by Γw(S) is defined as an undirected simple graph whose vertices are S and two distinct vertices x and y are adjacent if and only if xy 2 N(S), where S= Sn f0g and N(S) is the set of all non-zero nilpotent elements of S. In this paper, we determine the diameter of weakly nilpotent graph of an Artinian semiring. We prove that if w(S) is a forest, then Γw(S) is a union of a star and some isolated vertices. We study the clique number, the chromatic number and the independence number of Γw(S). Among other results, we show that for an Artinian semiring S, Γw(S) is not a disjoint union of cycles or a unicyclic graph. For Artinian semirings, we determine diam(Γw(S)). Finally, we characterize all commutative semirings S for which Γw(S) is a cycle, where w(S) is the complement of the weakly nilpotent graph of S. Finally, we characterize all commutative semirings S for which Γw(S) is a cycle.

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Let R be a ring with unity. Taloukolaei and Sahebi [2] introduced the Von Neumann regular graph GV nr+(R) of a ring, whose vertex set is R and two distinct vertices x and y are adjacent if and only if x + y is a Von Neumann regular element. In this article, we investigate some new properties of GV nr+(R) such as traversability, pancyclic, unicyclic, chordal and perfect. We also investigate the domination parameters of GV nr+(R) such as dominating set, domination number, total domination number, connected domination number and give the condition when the GV nr+(R) is an excellent graph. Finally we determine the bondage number.