Connectivity of the zero-divisor graph for finite rings

The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all non-zero (one-sided and two-sided) zero-divisors of R, and two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. [S. P. Redmond, The zero-divisor graph of a noncommutative ring, Int. J. Commut. Rings1(4) (2002) 203–211.] In the present paper, all finite rings with Eulerian zero-divisor graphs are described.

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ON VARIETIES OF RINGS WHOSE FINITE RINGS ARE DETERMINED BY THEIR ZERO-DIVISOR GRAPHS

Asian-European Journal of Mathematics ◽

10.1142/s1793557112500192 ◽

2012 ◽

Vol 05 (02) ◽

pp. 1250019 ◽

Cited By ~ 1

Author(s):

A. S. Kuzmina ◽

Yu. N. Maltsev

Keyword(s):

Associative Ring ◽

Zero Divisor ◽

Finite Ring ◽

Finite Rings ◽

Zero Divisor Graph ◽

Zero Divisors

The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all nonzero zero-divisors (one-sided and two-sided) of R, and two distinct vertices x and y are joined by an edge if and only if either xy = 0 or yx = 0. In the present paper, we study some properties of ring varieties where every finite ring is uniquely determined by its zero-divisor graph.

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Finite Rings Whose Graphs Have Clique Number Less than Five

Algebra Colloquium ◽

10.1142/s1005386721000419 ◽

2021 ◽

Vol 28 (03) ◽

pp. 533-540

Author(s):

Qiong Liu ◽

Tongsuo Wu ◽

Jin Guo

Keyword(s):

Commutative Ring ◽

Zero Divisor ◽

Clique Number ◽

Commutative Rings ◽

Finite Rings ◽

Zero Divisor Graph ◽

Finite Commutative Rings

Let [Formula: see text] be a commutative ring and [Formula: see text] be its zero-divisor graph. We completely determine the structure of all finite commutative rings whose zero-divisor graphs have clique number one, two, or three. Furthermore, if [Formula: see text] (each [Formula: see text] is local for [Formula: see text]), we also give algebraic characterizations of the ring [Formula: see text] when the clique number of [Formula: see text] is four.

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NILPOTENT FINITE RINGS WITH PLANAR ZERO-DIVISOR GRAPHS

Asian-European Journal of Mathematics ◽

10.1142/s179355710800045x ◽

2008 ◽

Vol 01 (04) ◽

pp. 565-574 ◽

Cited By ~ 9

Author(s):

A. S. KUZ'MINA ◽

Yu. N. MALTSEV

Keyword(s):

Associative Ring ◽

Zero Divisor ◽

Finite Rings ◽

Zero Divisor Graph ◽

Zero Divisors

The zero-divisor graph Γ(R) of an associative ring R is the graph with all vertices non-zero zero-divisors (one-sided and two-sided) of R, and two distinct vertices x and y are joined by an edge iff xy = 0 or yx = 0 ([10]). In the present paper, we describe all nilpotent finite rings with planar zero-divisor graphs.

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THE APPLICATIONS OF ZERO DIVISORS OF SOME FINITE RINGS OF MATRICES IN PROBABILITY AND GRAPH THEORY

Jurnal Teknologi ◽

10.11113/jurnalteknologi.v83.14936 ◽

2020 ◽

Vol 83 (1) ◽

pp. 127-132

Author(s):

Nurhidayah Zaid ◽

Nor Haniza Sarmin ◽

Sanhan Muhammad Salih Khasraw

Keyword(s):

Graph Theory ◽

Directed Graph ◽

Zero Divisor ◽

Finite Ring ◽

Finite Rings ◽

Zero Divisor Graph ◽

Zero Divisors ◽

Random Elements

Let R be a finite ring. The zero divisors of R are defined as two nonzero elements of R, say x and y where xy = 0. Meanwhile, the probability that two random elements in a group commute is called the commutativity degree of the group. Some generalizations of this concept have been done on various groups, but not in rings. In this study, a variant of probability in rings which is the probability that two elements of a finite ring have product zero is determined for some ring of matrices over integers modulo n. The results are then applied into graph theory, specifically the zero divisor graph. This graph is defined as a graph where its vertices are zero divisors of R and two distinct vertices x and y are adjacent if and only if xy = 0. It is found that the zero divisor graph of R is a directed graph.

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The zero divisor graphs of finite rings of cubefree order

Filomat ◽

10.2298/fil1508715t ◽

2015 ◽

Vol 29 (8) ◽

pp. 1715-1720 ◽

Cited By ~ 2

Author(s):

Adel Tadayyonfar ◽

Ali Ashrafi

Keyword(s):

Zero Divisor ◽

Finite Rings ◽

Zero Divisor Graph

The aim of this paper is to classify the zero divisor graph of finite rings of cubefree order. It is proved that all zero divisor graphs can be interpreted as the extended join over well-known graphs.

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