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
Chemical compounds are modeled as graphs.The atoms of molecules represent the graph vertices while chemical bonds between the atoms express the edges.The topological indices representing the molecular graph correspond...Chemical compounds are modeled as graphs.The atoms of molecules represent the graph vertices while chemical bonds between the atoms express the edges.The topological indices representing the molecular graph corresponds to the different chemical properties of compounds.Let a,b be are two positive integers,andΓ(Z_(a)×Z_(b))be the zero-divisor graph of the commutative ring Z_(a)×Z_(b).In this article some direct questions have been answered that can be utilized latterly in different applications.This study starts with simple computations,leading to a quite complex ring theoretic problems to prove certain properties.The theory of finite commutative rings is useful due to its different applications in the fields of advanced mechanics,communication theory,cryptography,combinatorics,algorithms analysis,and engineering.In this paper we determine the distance-based topological polynomials and indices of the zero-divisor graph of the commutative ring Z_(p^(2))×Z_(q)(for p,q as prime numbers)with the help of graphical structure analysis.The study outcomes help in understanding the fundamental relation between ring-theoretic and graph-theoretic properties of a zero-divisor graphΓ(G).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 tha...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 G = Γ(S) be a semigroup graph, i.e., a zero-divisor graph of a semigroup S with zero element 0. For any adjacent vertices x, y in G, denote C(x,y) = {z∈V(G) | N (z) = {x,y}}. Assume that in G there exi...Let G = Γ(S) be a semigroup graph, i.e., a zero-divisor graph of a semigroup S with zero element 0. For any adjacent vertices x, y in G, denote C(x,y) = {z∈V(G) | N (z) = {x,y}}. Assume that in G there exist two adjacent vertices x, y, a vertex s∈C(x,y) and a vertex z such that d (s,z) = 3. This paper studies algebraic properties of S with such graphs G = Γ(S), giving some sub-semigroups and ideals of S. It constructs some classes of such semigroup graphs and classifies all semigroup graphs with the property in two cases.展开更多
The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamilt...The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.展开更多
如果它是一张星图的精炼，一张图被称为一张星图的合适的精炼，但是它是既不一张星图也不一张完全的图。为有中心 c 的星图 G 的精炼，让 G c * 在顶点上导致的 G 的 subgraph 被给 V (G)\{ 邻近 c 的 c 或结束顶点 } 。在这份报纸，我...如果它是一张星图的精炼，一张图被称为一张星图的合适的精炼，但是它是既不一张星图也不一张完全的图。为有中心 c 的星图 G 的精炼，让 G c * 在顶点上导致的 G 的 subgraph 被给 V (G)\{ 邻近 c 的 c 或结束顶点 } 。在这份报纸，我们由调查他们的零除数的图 G =(R) 学习一些有限可交换的本地戒指 R 的同形的分类，它确切是一张星图与的合适的精炼一中心 c。我们决定所有有限可交换的本地人包围 R 以便 G c * 至少二连接了部件。我们证明导致的图 G c 的直径 * 如果，是二 Z (R) 2 { 0 } ， Z (R) 3 ={ 0 } 并且 G c * 被连接。我们决定有二个不同 nonadjacent 顶点的 R 的结构， Z (R)*\{ c } 以便理想[N()N()]{ 0 } 被仅仅 Z (R) 的一个元素产生 *\{ c } 。我们完全也与邻近 K n 的一个单个顶点的一些结束顶点决定在可交换的戒指和有限的完全的图 K n 之间的通讯。展开更多
Let Pn be a path graph with n vertices, and let Fn = Pn ∪ {c}, where c is adjacent to all vertices of Pn. The resulting graph is called a fan-shaped graph. The corresponding zero-divisor semigroups have been complete...Let Pn be a path graph with n vertices, and let Fn = Pn ∪ {c}, where c is adjacent to all vertices of Pn. The resulting graph is called a fan-shaped graph. The corresponding zero-divisor semigroups have been completely determined by Tang et al. for n = 2, 3, 4 and by Wu et al. for n ≥ 6, respectively. In this paper, we study the case for n = 5, and give all the corresponding zero-divisor semigroups of Fn.展开更多
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.
文摘Chemical compounds are modeled as graphs.The atoms of molecules represent the graph vertices while chemical bonds between the atoms express the edges.The topological indices representing the molecular graph corresponds to the different chemical properties of compounds.Let a,b be are two positive integers,andΓ(Z_(a)×Z_(b))be the zero-divisor graph of the commutative ring Z_(a)×Z_(b).In this article some direct questions have been answered that can be utilized latterly in different applications.This study starts with simple computations,leading to a quite complex ring theoretic problems to prove certain properties.The theory of finite commutative rings is useful due to its different applications in the fields of advanced mechanics,communication theory,cryptography,combinatorics,algorithms analysis,and engineering.In this paper we determine the distance-based topological polynomials and indices of the zero-divisor graph of the commutative ring Z_(p^(2))×Z_(q)(for p,q as prime numbers)with the help of graphical structure analysis.The study outcomes help in understanding the fundamental relation between ring-theoretic and graph-theoretic properties of a zero-divisor graphΓ(G).
基金Supported by Guangxi Natural Sciences Foundation(0575052,0640070)Supported byInnovation Project of Guangxi Graduate Education(2006106030701M05)Supported Scientific Research Foun-dation of Guangxi Educational Committee
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 G = Γ(S) be a semigroup graph, i.e., a zero-divisor graph of a semigroup S with zero element 0. For any adjacent vertices x, y in G, denote C(x,y) = {z∈V(G) | N (z) = {x,y}}. Assume that in G there exist two adjacent vertices x, y, a vertex s∈C(x,y) and a vertex z such that d (s,z) = 3. This paper studies algebraic properties of S with such graphs G = Γ(S), giving some sub-semigroups and ideals of S. It constructs some classes of such semigroup graphs and classifies all semigroup graphs with the property in two cases.
文摘The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.
基金Supported by National Natural Science Foundation of China （Grant No. 10671122） the first author is supported by Youth Foundation of Shanghai （Grant No. sdl10017） and also partly supported by Natural Science Foundation of Shanghai （Grant No. 10ZR1412500） the second author is partly supported by STCSM （Grant No. 09XD1402500）
文摘如果它是一张星图的精炼，一张图被称为一张星图的合适的精炼，但是它是既不一张星图也不一张完全的图。为有中心 c 的星图 G 的精炼，让 G c * 在顶点上导致的 G 的 subgraph 被给 V (G)\{ 邻近 c 的 c 或结束顶点 } 。在这份报纸，我们由调查他们的零除数的图 G =(R) 学习一些有限可交换的本地戒指 R 的同形的分类，它确切是一张星图与的合适的精炼一中心 c。我们决定所有有限可交换的本地人包围 R 以便 G c * 至少二连接了部件。我们证明导致的图 G c 的直径 * 如果，是二 Z (R) 2 { 0 } ， Z (R) 3 ={ 0 } 并且 G c * 被连接。我们决定有二个不同 nonadjacent 顶点的 R 的结构， Z (R)*\{ c } 以便理想[N()N()]{ 0 } 被仅仅 Z (R) 的一个元素产生 *\{ c } 。我们完全也与邻近 K n 的一个单个顶点的一些结束顶点决定在可交换的戒指和有限的完全的图 K n 之间的通讯。
基金Supported by the Guangxi Natural Science Foundation (Grant Nos.2010GXNSFB0130480991102+1 种基金2011GXNSFA018139)the Scientific Research Foundation of Guangxi Educational Committee (Grant No. 200911LX275)
文摘Let Pn be a path graph with n vertices, and let Fn = Pn ∪ {c}, where c is adjacent to all vertices of Pn. The resulting graph is called a fan-shaped graph. The corresponding zero-divisor semigroups have been completely determined by Tang et al. for n = 2, 3, 4 and by Wu et al. for n ≥ 6, respectively. In this paper, we study the case for n = 5, and give all the corresponding zero-divisor semigroups of Fn.
基金supported by the National Natural Science Foundation of China (10771095)the Guangxi Science Foundation(0832107,0991102)the Scientific Research Foundation of Guangxi Educational Committee (200707LX233)
基金Supported by National Natural Science Foundation(11161006)Guan-gxi Natural Science Foundation(2010GXNSFB013048,2011GXNSFA018139)Scientific Research Foundation of Guangxi Educational Committee(201012MS140)
