Journal ArticleDOI
On the total graph of a commutative ring without the zero element
David F. Anderson,Ayman Badawi +1 more
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In this article, the authors studied the two induced subgraphs Z0(Γ(R)) and T0( ΔΔ(R) of a commutative ring with nonzero identity, with vertices Z(R)\{0} and R\{0}, respectively.Abstract:
Let R be a commutative ring with nonzero identity, and let Z(R) be its set of zero-divisors. The total graph of R is the (undirected) graph T(Γ(R)) with vertices all elements of R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). In this paper, we study the two (induced) subgraphs Z0(Γ(R)) and T0(Γ(R)) of T(Γ(R)), with vertices Z(R)\{0} and R\{0}, respectively. We determine when Z0(Γ(R)) and T0(Γ(R)) are connected and compute their diameter and girth. We also investigate zero-divisor paths and regular paths in T0(Γ(R)).read more
Citations
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The generalized total graph of a commutative ring
David F. Anderson,Ayman Badawi +1 more
TL;DR: In this paper, the generalized total graph of a commutative ring with nonzero identity (GTH(R)) was investigated and the structure of GTH (R) was investigated.
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On the dot product graph of a commutative ring
TL;DR: In this paper, it was shown that for a commutative ring A and n ≥ 3, the graph ZD(R) is connected with diameter two (at most three) and girth three.
Journal ArticleDOI
Some results on the intersection graphs of ideals of rings
TL;DR: In this article, it was shown that for a ring R, if the clique number of G(R) is finite, then the chromatic number is finite and if R is a reduced ring, then both are equal.
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The intersection graph of gamma sets in the total graph of a commutative ring-i
T. Tamizh Chelvam,T. Asir +1 more
TL;DR: Chelvam and Asir as discussed by the authors studied the intersection graph ITΓ(R) of gamma sets in a commutative Artin ring and obtained the domination number of the total graph.
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A graph associated with the set of all nonzero annihilating ideals of a commutative ring
S. Visweswaran,Hiren D. Patel +1 more
TL;DR: This paper considers a simple undirected graph associated with R denoted by Ω(R) whose vertex set equals the set of all nonzero annihilating ideals of R and two distinct vertices I, J in this graph are joined by an edge if and only if I + J is also an annihilating ideal of R.
References
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Book
Modern graph theory
TL;DR: This book presents an account of newer topics, including Szemer'edi's Regularity Lemma and its use; Shelah's extension of the Hales-Jewett Theorem; the precise nature of the phase transition in a random graph process; the connection between electrical networks and random walks on graphs; and the Tutte polynomial and its cousins in knot theory.
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The Zero-Divisor Graph of a Commutative Ring☆
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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Coloring of commutative rings
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.
Journal ArticleDOI
The total graph of a commutative ring
David F. Anderson,Ayman Badawi +1 more
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.