# Some properties and domination number of the complement of a new graph associated to a commutative ring

01 Jan 2018-Vol. 3, Iss: 12, pp 99-108

TL;DR: It is shown that ΩR∗ is a connected graph with diam(ΩR ∼3) ≤ 3 unless R is isomorphic to a direct product of two fields and all commutative rings R with at least two maximal ideals for which Ω R∗ are planar are characterized.

Abstract: In this paper some properties of the complement of a new graph associated with a commutative ring are investigated ....

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TL;DR: For each commutative ring R we associate a simple graph Γ(R) as discussed by the authors, and we investigate the interplay between the ring-theoretic properties of R and the graph-theory properties of Γ (R).

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TL;DR: In this article, the authors present the idea of coloring of a commutative ring and show that the existence of an infinite clique implies that the clique R = co implies that there exists an infinitely many cliques.

956 citations

### "Some properties and domination numb..." refers background in this paper

...Beck in [9], during the last two decades, several researchers have associated a graph with certain subsets of a ring and explored the interplay between the ring-theoretic properties of a ring...

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17 Dec 1999

TL;DR: In this article, direct-directed graphs are used to describe the properties of trees, independent sets and matchings, and Eulerian and Hamiltonian graphs for graph colouring.

Abstract: Basic Results.- Directed Graphs.- Connectivity.- Trees.- Independent Sets and Matchings.- Eulerian and Hamiltonian Graphs.- Graph Colourings.- Planarity.- Triangulated Graphs.- Applications.

448 citations

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TL;DR: In this paper, it was shown that if A is a regular Noetherian ring with maximal ideals N 1,..., Ns, such that each A/Ni is finite, then for R = A/Nn11 ··· Nnss, χ(R) = cl(R).

331 citations

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TL;DR: In this article, the authors introduced and investigated the total graph of R, denoted by T ( Γ ( R ) ), which is the (undirected) graph with all elements of R as vertices.

290 citations