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On directed zero-divisor graphs of finite rings
[article]

2004
*
arXiv
*
pre-print

For an artinian

arXiv:math/0403062v2
fatcat:qtnizmly4nagjatrs4yx4yjwgi
*ring*R, the*directed**zero*-*divisor**graph*Γ(R) is connected if and only if there is no proper*one*-sided identity element in R. ... Such a*ring*R is also the only*ring*such that Γ (R) has exactly*one*source. This shows that Γ(R) can not be a network for any*ring*R. ... In this paper, we study the*directed**zero*-*divisor**graph**of*noncommutative*rings*and we focus our attention*on**finite**rings*(Most results*on**finite**rings*in this paper actually holds for artinian*rings*. ...##
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On directed zero-divisor graphs of finite rings

2005
*
Discrete Mathematics
*

For an artinian

doi:10.1016/j.disc.2005.03.006
fatcat:q7nc4iswgfavfaglqx2gk4o3l4
*ring*R, the*directed**zero*-*divisor**graph*(R) is connected if and only if there is no proper*one*-sided identity element in R. ... Such a*ring*R is also the only*ring*such that (R) has exactly*one*source. This shows that (R) cannot be a network for any*finite*or infinite*ring*R. ... Acknowledgments The author expresses his sincere thanks to the anonymous referees for their careful reading, and for the helpful suggestions which greatly improved the exposition*of*the paper. ...##
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On the Adjacency Matrix and Neighborhood Associated with Zero-divisor Graph for Direct Product of Finite Commutative Rings

2013
*
International Journal of Computer Applications Technology and Research
*

The main purpose

doi:10.7753/ijcatr0203.1021
fatcat:2dyp3ckhsvfzdfsmg575fj63oa
*of*this paper is to study the*zero*-*divisor**graph*for*direct*product*of**finite*commutative*rings*. ... In our present investigation we discuss the*zero*-*divisor**graphs*for the following*direct*products:*direct*product*of*the*ring**of*integers under addition and multiplication modulo p and the*ring**of*integers ... Finally, we prove some results*of*annihilators*on**zero*-*divisor**graph**of*A B, for any two commutative*rings*A and B with unity. ...##
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Analysis of Adjacency Matrix and Neighborhood Associated with Zero Divisor Graph of Finite Commutative Rings

2011
*
International Journal of Computer Applications
*

In the first section

doi:10.5120/1822-2394
fatcat:ejpmwbhobvhxvmjyqcoyfqjcbq
*of*the paper, we study about neighborhood set*of*the*zero**divisor**graph**of**direct*product over*finite*commutative*rings*. ... The present investigation is concerned with*zero**divisor**graph**of**direct*Product*of**finite*commutative*rings*and to give some new ideas about its corresponding adjacency matrix. ... Finally we have proved some theorem*on*neighborhood*of*the*zero**divisor**graph**of**direct*product over*finite*commutative*rings*as well as for adjacency matrix. ...##
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THE APPLICATIONS OF ZERO DIVISORS OF SOME FINITE RINGS OF MATRICES IN PROBABILITY AND GRAPH THEORY

2020
*
Jurnal Teknologi
*

It is found that the

doi:10.11113/jurnalteknologi.v83.14936
fatcat:kgn7baxyordppg77ytyracdtmy
*zero**divisor**graph**of*R is a*directed**graph*. ... 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. ... The first author would also like to express her gratitude to UTM and the Ministry*of*Higher Education, Malaysia for the financial funding through UTM Zamalah Scholarship. ...##
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Finite rings with complete bipartite zero-divisor graphs

2012
*
Russian Mathematics (Izvestiya VUZ. Matematika)
*

We describe all

doi:10.3103/s1066369x12030048
fatcat:gdq56qzmgvbthai65mleizs36m
*finite*associative*rings*with complete bipartite*zero*-*divisor**graphs*. ... Papers [4] [5] [6] [7] completely describe*finite**rings*with planar*zero*-*divisor**graphs*. In [8]*one*gives a complete list*of**finite**rings*with Eulerian*zero*-*divisor**graphs*. ...*One**of*the main research*directions*in this field is the description*of**rings*with*zero*-*divisor**graphs*that satisfy certain conditions. ...##
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ZERO-DIVISOR GRAPHS OF FINITE DIRECT PRODUCTS OF FINITE NON-COMMUTATIVE RINGS AND SEMIGROUPS

English

2015
*
Journal of Algebra Number Theory Advances and Applications
*

English

We determine the number

doi:10.18642/jantaa_7100121521
fatcat:ydttqxke4jd3hgprfl57tug4u4
*of*edges*of*the*zero*-*divisor**graph**of*the*direct*product*of**finitely*many*finite*non-commutative*rings*or semigoups. ...*ZERO*-*DIVISOR**GRAPHS**OF**FINITE**DIRECT*… 3 In this paper, we determine a formula for the number*of*edges*of*the*zero*-*divisor**graph**of*a*direct*product*of*non-commutative*rings*or semigroups , 1 t R R × × ... The*Zero*-*Divisor**Graph**of*a*Direct*Product*of**Rings*In this section, we determine a formula for the number*of*edges in the*zero*-*divisor**graph**of*a*direct*product give a MatLab implementation*of*this latter ...##
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Graphs and Zero-Divisors

2010
*
The College Mathematics Journal
*

Recently, a new approach to studying the set

doi:10.4169/074683410x522008
fatcat:pwpfimcknzf6ljl3z6quu37m3y
*of**zero*-*divisors*has emerged from an unlikely*direction*:*graph*theory. ... In general, the set*of**zero*-*divisors*lacks algebraic structure. In particular, the set*of*all*zero*-*divisors**of*a*ring*R, denoted Z(R), is not always closed under addition. ...*On*the other hand, no*graph*Γ*of*diameter 7 is the*zero*-*divisor**graph**of*a commutative*ring*. ...##
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Eigenvalues of zero-divisor graphs of finite commutative rings

2020
*
Journal of Algebraic Combinatorics
*

AbstractWe investigate eigenvalues

doi:10.1007/s10801-020-00989-6
fatcat:rjb7kmje4vaxnaxmvcegqkgtge
*of*the*zero*-*divisor**graph*$$\Gamma (R)$$ Γ ( R )*of**finite*commutative*rings*R and study the interplay between these eigenvalues, the*ring*-theoretic properties*of*R and ... With this product, we find relations between the number*of*vertices*of*the*zero*-*divisor**graph*$$\Gamma (R)$$ Γ ( R ) , the compressed*zero*-*divisor**graph*, the structure*of*the*ring*R and the eigenvalues ... Definition 1.2 (*Zero*-*divisor**graph*) Let R be a*finite*commutative*ring*with 1 = 0 and let Z (R) denote its set*of**zero*-*divisors*. ...##
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Categorial properties of compressed zero-divisor graphs of finite commutative rings
[article]

2018
*
arXiv
*
pre-print

We define a compressed

arXiv:1807.11283v1
fatcat:suhy5txgsbfp5bx3o37v7whwia
*zero*-*divisor**graph*(K)*of*a*finite*commutative unital*ring*K, where the compression is performed by means*of*the associatedness relation. ... We use the structure*of*(K) to characterize important classes*of**finite*commutative unital*rings*, such as local*rings*and principal ideal*rings*. ... Clearly, Θ (0), the*graph**of*the*zero**ring*, is the*graph*with precisely*one*vertex and*one*loop. ...##
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Induced subgraphs of zero-divisor graphs
[article]

2022
*
arXiv
*
pre-print

The

arXiv:2207.11741v1
fatcat:ehrohkmdxrdevb4s3nwa6udna4
*zero*-*divisor**graph**of*a*finite*commutative*ring*with unity is the*graph*whose vertex set is the set*of**zero*-*divisors*in the*ring*, with a and b adjacent if ab=0. ... We show that the class*of**zero*-*divisor**graphs*is universal, in the sense that every*finite**graph*is isomorphic to an induced subgraph*of*a*zero*-*divisor**graph*. ... We are also grateful to the International Workshop*on**Graphs*from Algebraic Structures, organised by Manonmaniam Sundaranar University and the Academy*of*Discrete Mathematics and Applications, for the ...##
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Eigenvalues of zero-divisor graphs of finite commutative rings
[article]

2019
*
arXiv
*
pre-print

We investigate eigenvalues

arXiv:1910.12567v1
fatcat:lybj2vtxlfd6bddnb6en3at35a
*of*the*zero*-*divisor**graph*Γ(R)*of**finite*commutative*rings*R and study the interplay between these eigenvalues, the*ring*-theoretic properties*of*R and the*graph*-theoretic properties ... With this product, we find relations between the number*of*vertices*of*the*zero*-*divisor**graph*Γ(R), the compressed*zero*-*divisor**graph*, the structure*of*the*ring*R and the eigenvalues*of*Γ(R). ... For the set*of*units*of*a*ring*R, we write U (R). Products*of**zero*-*divisor**graphs*Let R ∼ = R 1 × . . . × R r be a*ring*, where each R i is a*finite*local*ring*. ...##
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Zero-Divisor Graphs and Zero-Divisor Functors
[article]

2022
*
arXiv
*
pre-print

Stopar, we introduce a new definition

arXiv:2203.02952v1
fatcat:xewu35te2fbzrac7bktcrad7me
*of**zero*-*divisor**graphs*attached to*rings*, that includes all*of*the classical definitions already known in the literature. ... We thus recover and generalize many known results for*zero*-*divisor**graphs*and provide a framework which might be useful for further investigations*on*this topic. ... There we show under which conditions*on*{R A } the decomposition*of*Z(A, R A ) as a*finite**direct*product*of**graphs*leads to a decomposition*of*A as a*finite**direct*product*of**rings*. ...##
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Zero-Divisor Graphs of Matrices Over Commutative Rings

2009
*
Communications in Algebra
*

We investigate the properties

doi:10.1080/00927870802465951
fatcat:yakja2bqobasjdcd2qodpr7h74
*of*(*directed*)*zero*-*divisor**graphs**of*matrix*rings*. ... Then we use these results to discuss the relation between the diameter*of*the*zero*-*divisor**graph**of*a commutative*ring*R and that*of*the matrix*ring*M n R . ... This work was partially supported by the Ministry*of*Science and Environmental Protection*of*the Republic*of*Serbia Project #144020. ...##
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On the zero divisor graphs of the ring of Lipschitz integers modulo n
[article]

2015
*
arXiv
*
pre-print

This article studies the

arXiv:1505.00102v1
fatcat:n3qbpxak75ht7aphmwyajoqe7m
*zero**divisor**graphs**of*the*ring**of*Lipschitz integers modulo n. In particular we focus*on*the number*of*vertices, the diameter and the girth. ... We also give some results regarding the domination number*of*these*graphs*. ... The*directed**zero**divisor**graph*, Γ(R), and the undirected*zero**divisor**graph*, Γ(R). Both*graphs*share the same vertex set, namely, the set Z(R) **of*non-*zero**zero**divisors**of*R. ...
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