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
For every simple graph G,a class of multiple clique cluster-whiskered graphs G^(eπm)is introduced,and it is shown that all such graphs are vertex decomposable;thus,the independence simplicial complex Ind G^(eπm)is s...For every simple graph G,a class of multiple clique cluster-whiskered graphs G^(eπm)is introduced,and it is shown that all such graphs are vertex decomposable;thus,the independence simplicial complex Ind G^(eπm)is sequentially Cohen-Macaulay.The properties of the graphs G^(eπm)and G^(π)constructed by Cook and Nagel are studied,including the enumeration of facets of the complex Ind G^(π)and the calculation of Betti numbers of the cover ideal Ic(G^(eπm).We also prove that the complex△=IndH is strongly shellable and pure for either a Boolean graph H=Bn or the full clique-whiskered graph H=G^(W)of C,which is obtained by adding a whisker to each vertex of G.This implies that both the facet ideal I(△)and the cover ideal Ic(H)have linear quotients.展开更多
We prove that if G is a gap-free and chair-free simple graph,then the regularity of the edge ideal of G is no more than 3.If G is a gap-free and P4-free graph,then it is a chair-free graph;furthermore,the complement o...We prove that if G is a gap-free and chair-free simple graph,then the regularity of the edge ideal of G is no more than 3.If G is a gap-free and P4-free graph,then it is a chair-free graph;furthermore,the complement of G is chordal,and thus the regularity of G is 2.展开更多
Let R be a commutative ring and Γ(R)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 R■R1&#...Let R be a commutative ring and Γ(R)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 R■R1×R2×…×Rn(each Ri is local for i=1,2,3,...,n),we also give algebraic characterizations of the ring R when the clique number of Γ(R)is four.展开更多
在这份报纸，我们介绍象 U 那样的一些新定义 * L * 描述 poset P 的零除数的图 G=(P) ，并且把一个新、快的证明给主要结果在里面的状况[2， 4 ] 。由与最少的度删除一个典型顶点，我们为发现有限的图 G 的一个最大的派系提供一个算法...在这份报纸，我们介绍象 U 那样的一些新定义 * L * 描述 poset P 的零除数的图 G=(P) ，并且把一个新、快的证明给主要结果在里面的状况[2， 4 ] 。由与最少的度删除一个典型顶点，我们为发现有限的图 G 的一个最大的派系提供一个算法。我们有关直径和尺寸学习 posets 的零除数的图的一些性质。我们也提供 posets 的成层的演讲。展开更多
semiring 是类似于一枚戒指的代数学的结构，但是没有要求，那个各个元素必须有添加剂逆。围住的 semiring 是与一份兼容围住的部分订单装备的 semiring。在这份报纸，零个除数的性质和围住的 semiring 的主要元素被学习。特别地，在一...semiring 是类似于一枚戒指的代数学的结构，但是没有要求，那个各个元素必须有添加剂逆。围住的 semiring 是与一份兼容围住的部分订单装备的 semiring。在这份报纸，零个除数的性质和围住的 semiring 的主要元素被学习。特别地，在一些温和假设下面， A 的非零零除数的集合 Z (A) 是 \ ，这被证明 { 0， 1 } ，并且 A 的每个主要元素是一个最大的元素。为有 Z (A)= 的围住的 semiring A \{ 0， 1 } ，如果 ACC 为 A 的元素或为歼灭 A 的理想的主管成立， A 有限地有许多最大的元素，这被证明如果 ACC 为 A 的元素或为歼灭 A 的理想的主管成立。作为主要元素的应用，我们证明围住的 semiring A 的结构被不可分的围住的 semirings 的结构完全决定是否任何一个 | Z (A)|= 1 或 | Z (A)|= 2 并且 Z (A)< 啜 class= “ a-plus-plus ” > 2 </sup> 0。到可交换的戒指的理想的结构的应用也被考虑。特别地，当 R 有理想的一个有限数字时， poset II (R) 的链建筑群纯、可轰炸，这被显示出，在 II (R) 由 R 的所有理想组成的地方。展开更多
We study the algebraic structure of rings R whose zero-divisor graph T(R)has clique number four.Furthermore,we give complete characterizations of all the finite commutative local rings with clique number 4.
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.
基金Supported by the Natural Science Foundation of Shanghai(No.19ZR1424100)the National Natural Science Foundation of China(No.11271250,11971338).
文摘For every simple graph G,a class of multiple clique cluster-whiskered graphs G^(eπm)is introduced,and it is shown that all such graphs are vertex decomposable;thus,the independence simplicial complex Ind G^(eπm)is sequentially Cohen-Macaulay.The properties of the graphs G^(eπm)and G^(π)constructed by Cook and Nagel are studied,including the enumeration of facets of the complex Ind G^(π)and the calculation of Betti numbers of the cover ideal Ic(G^(eπm).We also prove that the complex△=IndH is strongly shellable and pure for either a Boolean graph H=Bn or the full clique-whiskered graph H=G^(W)of C,which is obtained by adding a whisker to each vertex of G.This implies that both the facet ideal I(△)and the cover ideal Ic(H)have linear quotients.
基金Research supported by the Natural Science Foundation of Shanghai(No.19ZR1424100)the National Natural Science Foundation of China(No.11971338).
文摘We prove that if G is a gap-free and chair-free simple graph,then the regularity of the edge ideal of G is no more than 3.If G is a gap-free and P4-free graph,then it is a chair-free graph;furthermore,the complement of G is chordal,and thus the regularity of G is 2.
基金This research was supported by the National Natural Science Foundation of China(No.11801356,No.11401368,No.11971338)by the Natural Science Foundation of Shanghai(No.19ZR1424100).
文摘Let R be a commutative ring and Γ(R)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 R■R1×R2×…×Rn(each Ri is local for i=1,2,3,...,n),we also give algebraic characterizations of the ring R when the clique number of Γ(R)is four.
基金Supported by the National Natural Science Foundation of China （11271250）.Acknowledgements. The authors express their sincere thanks to the referees for the careful reading and suggestions which improved the exposition of the paper.
文摘在这份报纸，我们介绍象 U 那样的一些新定义 * L * 描述 poset P 的零除数的图 G=(P) ，并且把一个新、快的证明给主要结果在里面的状况[2， 4 ] 。由与最少的度删除一个典型顶点，我们为发现有限的图 G 的一个最大的派系提供一个算法。我们有关直径和尺寸学习 posets 的零除数的图的一些性质。我们也提供 posets 的成层的演讲。
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant No. 11271250）.
文摘semiring 是类似于一枚戒指的代数学的结构，但是没有要求，那个各个元素必须有添加剂逆。围住的 semiring 是与一份兼容围住的部分订单装备的 semiring。在这份报纸，零个除数的性质和围住的 semiring 的主要元素被学习。特别地，在一些温和假设下面， A 的非零零除数的集合 Z (A) 是 \ ，这被证明 { 0， 1 } ，并且 A 的每个主要元素是一个最大的元素。为有 Z (A)= 的围住的 semiring A \{ 0， 1 } ，如果 ACC 为 A 的元素或为歼灭 A 的理想的主管成立， A 有限地有许多最大的元素，这被证明如果 ACC 为 A 的元素或为歼灭 A 的理想的主管成立。作为主要元素的应用，我们证明围住的 semiring A 的结构被不可分的围住的 semirings 的结构完全决定是否任何一个 | Z (A)|= 1 或 | Z (A)|= 2 并且 Z (A)< 啜 class= “ a-plus-plus ” > 2 </sup> 0。到可交换的戒指的理想的结构的应用也被考虑。特别地，当 R 有理想的一个有限数字时， poset II (R) 的链建筑群纯、可轰炸，这被显示出，在 II (R) 由 R 的所有理想组成的地方。
基金The first author is supported by Fundamental Research Funds for the Central Universi- ties （No. XDJK2013C060）, Chongqing Research Program of Application Foundation and Advanced Technology （No. cstc2014jcyjA00028） and Scientific Research Foundation for Doctors of Southwest University （No. SWUl12054）. The second author is supported by National Natural Science Foundation of China （No. 11271250）.
基金This research was supported by the National Natural Science Foundation of China(No.11801356,No.11401368,No.11971338)by the Natural Science Foundation of Shanghai(No.19ZR1424100).
文摘We study the algebraic structure of rings R whose zero-divisor graph T(R)has clique number four.Furthermore,we give complete characterizations of all the finite commutative local rings with clique number 4.
