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

Tongsuo Wu
2004 arXiv   pre-print
For an artinian 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.  ... 
arXiv:math/0403062v2 fatcat:qtnizmly4nagjatrs4yx4yjwgi

On directed zero-divisor graphs of finite rings

Tongsuo Wu
2005 Discrete Mathematics  
For an artinian 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.  ... 
doi:10.1016/j.disc.2005.03.006 fatcat:q7nc4iswgfavfaglqx2gk4o3l4

On the Adjacency Matrix and Neighborhood Associated with Zero-divisor Graph for Direct Product of Finite Commutative Rings

Kuntala Patra, Priyanka Pratim Baruah
2013 International Journal of Computer Applications Technology and Research  
The main purpose 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.  ... 
doi:10.7753/ijcatr0203.1021 fatcat:2dyp3ckhsvfzdfsmg575fj63oa

Analysis of Adjacency Matrix and Neighborhood Associated with Zero Divisor Graph of Finite Commutative Rings

Pranjali Sharma, Amit Sharma, Deepak Chaudhr, R.K. Vats
2011 International Journal of Computer Applications  
In the first section 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.  ... 
doi:10.5120/1822-2394 fatcat:ejpmwbhobvhxvmjyqcoyfqjcbq

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

Nurhidayah Zaid, Nor Haniza Sarmin, Sanhan Muhammad Salih Khasraw
2020 Jurnal Teknologi  
It is found that the 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.  ... 
doi:10.11113/jurnalteknologi.v83.14936 fatcat:kgn7baxyordppg77ytyracdtmy

Finite rings with complete bipartite zero-divisor graphs

A. S. Kuzmina, Yu. N. Maltsev
2012 Russian Mathematics (Izvestiya VUZ. Matematika)  
We describe all 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.  ... 
doi:10.3103/s1066369x12030048 fatcat:gdq56qzmgvbthai65mleizs36m

ZERO-DIVISOR GRAPHS OF FINITE DIRECT PRODUCTS OF FINITE NON-COMMUTATIVE RINGS AND SEMIGROUPS
English

2015 Journal of Algebra Number Theory Advances and Applications  
We determine the number 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  ... 
doi:10.18642/jantaa_7100121521 fatcat:ydttqxke4jd3hgprfl57tug4u4

Graphs and Zero-Divisors

M. Axtell, J. Stickles
2010 The College Mathematics Journal  
Recently, a new approach to studying the set 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.  ... 
doi:10.4169/074683410x522008 fatcat:pwpfimcknzf6ljl3z6quu37m3y

Eigenvalues of zero-divisor graphs of finite commutative rings

Katja Mönius
2020 Journal of Algebraic Combinatorics  
AbstractWe investigate eigenvalues 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.  ... 
doi:10.1007/s10801-020-00989-6 fatcat:rjb7kmje4vaxnaxmvcegqkgtge

Categorial properties of compressed zero-divisor graphs of finite commutative rings [article]

Alen urić, Sara Jevenić, Nik Stopar
2018 arXiv   pre-print
We define a compressed 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.  ... 
arXiv:1807.11283v1 fatcat:suhy5txgsbfp5bx3o37v7whwia

Induced subgraphs of zero-divisor graphs [article]

G. Arunkumar, Peter J. Cameron, T. Kavaskar, T. Tamizh Chelvam
2022 arXiv   pre-print
The 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  ... 
arXiv:2207.11741v1 fatcat:ehrohkmdxrdevb4s3nwa6udna4

Eigenvalues of zero-divisor graphs of finite commutative rings [article]

Katja Mönius
2019 arXiv   pre-print
We investigate eigenvalues 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.  ... 
arXiv:1910.12567v1 fatcat:lybj2vtxlfd6bddnb6en3at35a

Zero-Divisor Graphs and Zero-Divisor Functors [article]

Enrico Sbarra, Maurizio Zanardo
2022 arXiv   pre-print
Stopar, we introduce a new definition 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.  ... 
arXiv:2203.02952v1 fatcat:xewu35te2fbzrac7bktcrad7me

Zero-Divisor Graphs of Matrices Over Commutative Rings

Ivana Božić, Zoran Petrović
2009 Communications in Algebra  
We investigate the properties 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.  ... 
doi:10.1080/00927870802465951 fatcat:yakja2bqobasjdcd2qodpr7h74

On the zero divisor graphs of the ring of Lipschitz integers modulo n [article]

Jose Maria Grau, Celino Miquel, Antonio Oller-Marcen
2015 arXiv   pre-print
This article studies the 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.  ... 
arXiv:1505.00102v1 fatcat:n3qbpxak75ht7aphmwyajoqe7m
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