The commuting graph of an arbitrary ring R, denoted by Г（R）, is a graph whose vertices are all non-central elements of R, and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we inv...The commuting graph of an arbitrary ring R, denoted by Г（R）, is a graph whose vertices are all non-central elements of R, and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we investigate the connectivity and the diameter of Г（ZnS3）. We show that Г（ZnS3） is connected if and only if n is not a prime number. If Г（ZnS3） is connected then diam（Г（ZnS3）） = 3, while ifГ（ZnS3） is disconnected then every connected component of Г（ZnS3） must be a complete graph with same size, and we completely determine the vertice set of every connected component.展开更多
In [1], Joe Warfel investigated the diameter of a zero-divisor graph for a direct product R 1 × R 2 with respect to the diameter of the zero-divisor graph of R 1 and R 2 . But the author only considered those gra...In [1], Joe Warfel investigated the diameter of a zero-divisor graph for a direct product R 1 × R 2 with respect to the diameter of the zero-divisor graph of R 1 and R 2 . But the author only considered those graphs whose diameters ≥ 1 and discussed six cases. This paper further discusses the other nine cases and also gives a complete characterization for the possible diameters for left Artin rings.展开更多
In this paper, a new class of rings, called FIC rings, is introduced for studying quasi-zero-divisor graphs of rings. Let R be a ring. The quasi-zero-divisor graph of R, denoted by Г＊（R）, is a directed graph defin...In this paper, a new class of rings, called FIC rings, is introduced for studying quasi-zero-divisor graphs of rings. Let R be a ring. The quasi-zero-divisor graph of R, denoted by Г＊（R）, is a directed graph defined on its nonzero quasi-zero-divisors, where there is an arc from a vertex x to another vertex y if and only if xRy = 0. We show that the following three conditions on an FIC ring R are equivalent： （1） χ（R） is finite; （2） ω（R） is finite; （3） Nil＊ R is finite where Nil.R equals the finite intersection of prime ideals. Furthermore, we also completely determine the connectedness, the diameter and the girth of Г＊ （R）.展开更多
This research investigates the comparative efficacy of generating zero divisor graphs (ZDGs) of the ring of integers ℤ<sub>n</sub> modulo n using MAPLE algorithm. Zero divisor graphs, pivotal in the study ...This research investigates the comparative efficacy of generating zero divisor graphs (ZDGs) of the ring of integers ℤ<sub>n</sub> modulo n using MAPLE algorithm. Zero divisor graphs, pivotal in the study of ring theory, depict relationships between elements of a ring that multiply to zero. The paper explores the development and implementation of algorithms in MAPLE for constructing these ZDGs. The comparative study aims to discern the strengths, limitations, and computational efficiency of different MAPLE algorithms for creating zero divisor graphs offering insights for mathematicians, researchers, and computational enthusiasts involved in ring theory and mathematical computations.展开更多
We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that...We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that the maximal right quotient ring of a potent semiprimitive normal ring is abelian regular, moreover, the zero-divisor graph of such a ring is studied.展开更多
Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a an...Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in are adjacent if and only if and . In this paper, we investigate some combinatorial properties of the cozero-divisor graphs and such as connectivity, diameter, girth, clique numbers and planarity. We also study the cozero-divisor graphs of the direct products of two arbitrary commutative rings.展开更多
Let R be a commutative ring with identity and M an R-module. In this paper, we relate a graph to M, say Γ(M), provided tsshat when M=R, Γ(M)is exactly the classic zero-divisor graph.
We introduce Tribonacci cordial labeling as an extension of Fibonacci cordial labeling, a well-known form of vertex-labelings. A graph that admits Tribonacci cordial labeling is called Tribonacci cordial graph. In thi...We introduce Tribonacci cordial labeling as an extension of Fibonacci cordial labeling, a well-known form of vertex-labelings. A graph that admits Tribonacci cordial labeling is called Tribonacci cordial graph. In this paper we investigate whether some well-known graphs are Tribonacci cordial.展开更多
This paper investigates the connections between ring theory, module theory, and graph theory through the graph G(R)of a ring R. We establish that vertices of G(R)correspond to modules, with edges defined by the vanish...This paper investigates the connections between ring theory, module theory, and graph theory through the graph G(R)of a ring R. We establish that vertices of G(R)correspond to modules, with edges defined by the vanishing of their tensor product. Key results include the graph’s connectivity, a diameter of at most 3, and a girth of at most 7 when cycles are present. We show that the set of modules S(R)is empty if and only if R is a field, and that for semisimple rings, the diameter is at most 2. The paper also discusses module isomorphisms over subrings and localization, as well as the inclusion of G(T)within G(R)for a quotient ring T, highlighting that the reverse inclusion is not guaranteed. Finally, we provide an example illustrating that a non-finitely generated module M does not imply M⊗M=0. These findings deepen our understanding of the interplay among rings, modules, and graphs.展开更多
通过中国知网(CNKI)与Web of Science(WOS)核心数据库,检索其中2004年~2024年所收录的环锭包芯纱相关中英文文献,采集相关文献数据。利用CiteSpace知识图谱工具,构建环锭包芯纱研究的知识图谱,对环锭包芯纱领域的年度发文量、作者和国...通过中国知网(CNKI)与Web of Science(WOS)核心数据库,检索其中2004年~2024年所收录的环锭包芯纱相关中英文文献,采集相关文献数据。利用CiteSpace知识图谱工具,构建环锭包芯纱研究的知识图谱,对环锭包芯纱领域的年度发文量、作者和国家合作分布、研究热点与发展趋势进行可视化分析。研究发现:国内环锭包芯纱领域年度发文量总体上呈现逐年递减的趋势;国际年度发文量变化幅度较大,整体上是一个增长的态势。展开更多
图是一种非常重要的数据结构形式,被广泛用于社交网络、交通网络和搜索引擎等领域。随着图数据规模爆发式增长,存储容量受限,分布式图计算成为处理大规模图数据的焦点。宽度优先搜索(breadth first search,BFS)算法是图遍历和许多图分...图是一种非常重要的数据结构形式,被广泛用于社交网络、交通网络和搜索引擎等领域。随着图数据规模爆发式增长,存储容量受限,分布式图计算成为处理大规模图数据的焦点。宽度优先搜索(breadth first search,BFS)算法是图遍历和许多图分析算法的基础,而在分布式图计算过程中存在严重的通信开销。针对上述问题,本文提出了一种综合的数据压缩编码优化方案,结合位图和变长压缩数组,通过更高的压缩率来降低数据通信开销;此外,还提出了一种点对点异步环形通信策略,进一步降低分布式图计算中计算-通信的同步开销。通过这些优化手段,本文在8节点的分布式集群上对优化后BFS算法的性能进行了系统评估,结果表明,当图数据规模为28时,优化后的BFS算法平均性能为46.79亿条边每秒遍历(giga-traversed edges per second,GTEPS),性能比优化前提升了接近7.82%。展开更多
基金The NSF(10971024)of Chinathe Specialized Research Fund(200802860024)for the Doctoral Program of Higher Educationthe NSF(BK2010393)of Jiangsu Province
文摘The commuting graph of an arbitrary ring R, denoted by Г（R）, is a graph whose vertices are all non-central elements of R, and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we investigate the connectivity and the diameter of Г（ZnS3）. We show that Г（ZnS3） is connected if and only if n is not a prime number. If Г（ZnS3） is connected then diam（Г（ZnS3）） = 3, while ifГ（ZnS3） is disconnected then every connected component of Г（ZnS3） must be a complete graph with same size, and we completely determine the vertice set of every connected component.
基金Supported by the Natural Sciences Foundation of Guangxi Province(0575052, 0640070)Supported by the Innovation Project of Guangxi Graduate Education(2006106030701M05)Supported by the Scientific Research Foundation of Guangxi Educational Committee(200707LX233
文摘In [1], Joe Warfel investigated the diameter of a zero-divisor graph for a direct product R 1 × R 2 with respect to the diameter of the zero-divisor graph of R 1 and R 2 . But the author only considered those graphs whose diameters ≥ 1 and discussed six cases. This paper further discusses the other nine cases and also gives a complete characterization for the possible diameters for left Artin rings.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1137134311161006+4 种基金1166101411171142)the Guangxi Science Research and Technology Development Project(Grant No.1599005-2-13)the Scientic Research Fund of Guangxi Education Department(Grant No.KY2015ZD075)the Natural Science Foundation of Guangxi(Grant No.2016GXSFDA380017)
文摘In this paper, a new class of rings, called FIC rings, is introduced for studying quasi-zero-divisor graphs of rings. Let R be a ring. The quasi-zero-divisor graph of R, denoted by Г＊（R）, is a directed graph defined on its nonzero quasi-zero-divisors, where there is an arc from a vertex x to another vertex y if and only if xRy = 0. We show that the following three conditions on an FIC ring R are equivalent： （1） χ（R） is finite; （2） ω（R） is finite; （3） Nil＊ R is finite where Nil.R equals the finite intersection of prime ideals. Furthermore, we also completely determine the connectedness, the diameter and the girth of Г＊ （R）.
文摘This research investigates the comparative efficacy of generating zero divisor graphs (ZDGs) of the ring of integers ℤ<sub>n</sub> modulo n using MAPLE algorithm. Zero divisor graphs, pivotal in the study of ring theory, depict relationships between elements of a ring that multiply to zero. The paper explores the development and implementation of algorithms in MAPLE for constructing these ZDGs. The comparative study aims to discern the strengths, limitations, and computational efficiency of different MAPLE algorithms for creating zero divisor graphs offering insights for mathematicians, researchers, and computational enthusiasts involved in ring theory and mathematical computations.
基金Partially supported by the NSF (10071035) of China.
文摘We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that the maximal right quotient ring of a potent semiprimitive normal ring is abelian regular, moreover, the zero-divisor graph of such a ring is studied.
文摘Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in are adjacent if and only if and . In this paper, we investigate some combinatorial properties of the cozero-divisor graphs and such as connectivity, diameter, girth, clique numbers and planarity. We also study the cozero-divisor graphs of the direct products of two arbitrary commutative rings.
文摘Let R be a commutative ring with identity and M an R-module. In this paper, we relate a graph to M, say Γ(M), provided tsshat when M=R, Γ(M)is exactly the classic zero-divisor graph.
文摘We introduce Tribonacci cordial labeling as an extension of Fibonacci cordial labeling, a well-known form of vertex-labelings. A graph that admits Tribonacci cordial labeling is called Tribonacci cordial graph. In this paper we investigate whether some well-known graphs are Tribonacci cordial.
文摘This paper investigates the connections between ring theory, module theory, and graph theory through the graph G(R)of a ring R. We establish that vertices of G(R)correspond to modules, with edges defined by the vanishing of their tensor product. Key results include the graph’s connectivity, a diameter of at most 3, and a girth of at most 7 when cycles are present. We show that the set of modules S(R)is empty if and only if R is a field, and that for semisimple rings, the diameter is at most 2. The paper also discusses module isomorphisms over subrings and localization, as well as the inclusion of G(T)within G(R)for a quotient ring T, highlighting that the reverse inclusion is not guaranteed. Finally, we provide an example illustrating that a non-finitely generated module M does not imply M⊗M=0. These findings deepen our understanding of the interplay among rings, modules, and graphs.
文摘通过中国知网(CNKI)与Web of Science(WOS)核心数据库,检索其中2004年~2024年所收录的环锭包芯纱相关中英文文献,采集相关文献数据。利用CiteSpace知识图谱工具,构建环锭包芯纱研究的知识图谱,对环锭包芯纱领域的年度发文量、作者和国家合作分布、研究热点与发展趋势进行可视化分析。研究发现:国内环锭包芯纱领域年度发文量总体上呈现逐年递减的趋势;国际年度发文量变化幅度较大,整体上是一个增长的态势。
文摘图是一种非常重要的数据结构形式,被广泛用于社交网络、交通网络和搜索引擎等领域。随着图数据规模爆发式增长,存储容量受限,分布式图计算成为处理大规模图数据的焦点。宽度优先搜索(breadth first search,BFS)算法是图遍历和许多图分析算法的基础,而在分布式图计算过程中存在严重的通信开销。针对上述问题,本文提出了一种综合的数据压缩编码优化方案,结合位图和变长压缩数组,通过更高的压缩率来降低数据通信开销;此外,还提出了一种点对点异步环形通信策略,进一步降低分布式图计算中计算-通信的同步开销。通过这些优化手段,本文在8节点的分布式集群上对优化后BFS算法的性能进行了系统评估,结果表明,当图数据规模为28时,优化后的BFS算法平均性能为46.79亿条边每秒遍历(giga-traversed edges per second,GTEPS),性能比优化前提升了接近7.82%。
