Author

# Philip S. Livingston

Bio: Philip S. Livingston is an academic researcher from University of Tennessee. The author has contributed to research in topics: Commutative ring & Zero divisor. The author has an hindex of 2, co-authored 2 publications receiving 957 citations.

##### Papers

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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).

1,087 citations

##### Cited by

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TL;DR: In this paper, it was shown that if G and H are two non-abelian finite groups such that Γ G ≅ Γ H, then | G | = | H |, then H is nilpotent.

304 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

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TL;DR: In this paper, an undirected graph Γ(S) is associated to each commutative multiplicative semigroup S with 0, where vertices of the graph are labeled by the nonzero zero-divisors of S, and two vertices x,y are connected by an edge in case xy = 0 in S.

Abstract: An undirected graph Γ(S) is associated to each commutative multiplicative semigroup S with 0. The vertices of the graph are labeled by the nonzero zero-divisors of S , and two vertices x,y are connected by an edge in case xy = 0 in S . The properties and possible structures of the graph Γ (S) are studied.

214 citations

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TL;DR: In this paper, a natural graph associated to the zero-divisors of a commutative ring is considered and the cycle-structure of this graph is classified and some group-theoretic properties of the group of graph-automorphisms are established.

Abstract: There is a natural graph associated to the zero-divisors of a commutative ring In this article we essentially classify the cycle-structure of this graph and establish some group-theoretic properties of the group of graph-automorphisms We also determine the kernel of the canonical homomorphism from to

195 citations

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TL;DR: For a commutative ring R with set of zero-divisors Z (R), the zero-Divisor graph of R is Γ( R ) = Z ( R )−{0), with distinct vertices x and y adjacent if and only if xy = 0 as mentioned in this paper.

194 citations