关于二次型图的完美性

本文应用强完美图定理,解决了二次型图的完美图判别问题.     

完美路图P3（G）

P_k（G）是指这样的图：G中的所有k路作为P_k（G）的顶点集，两个不同的顶点在Pk（G）中邻接当且仅当它们所对应的两条k路的并为G中的（k＋1）路或k圈，那么，完美图猜想对于路图P_3（G）是成立的．     

强完美图定理及相关的问题

介绍强完美图定理(The Strong Perfect Graph Theorem,SPGT)的历史和获证经过,同时简述SPGT被克服后生发的一些新问题,以期对图理论的一般研究起到鼓励和促进作用.因具体的证明浩大聱曲,在技术的部分仅注重框架而不涉及细节.     

图^nUi=1 Fmi,t和^nUi=1 Hmi,t的优美性

本文构造了两类非连通图^nUi=1 Fmi,t和^nUi=1 Hmi,t并证明了这两类图是优美的，且也是交错的。     

紧 n -Lie 代数与不变双线性型

该文研究 n -Lie代数的非退化不变双线性型. 给出了实数域上紧 n -Lie 代数的概念, 并对其结构进行了研究.     

图U(F_(m_i,t)) from i=1 to n和U(F_(m_i,t)) from i=1 to n的优美性

本文构造了两类非连通图U(F_(m_i,t)) from i=1 to n和U(F_(m_i,t)) from i=1 to n,并证明了这两类图是优美的,且也是交错的.     

图Cn及其r-冠的新的优美标号

研究了关于图的r-冠的优美标号的一个问题,证明了：当n≡0,3（mod 4）时,图Cn及其r-冠是优美图,所给出的新的优美标号不同于现有文献中得到的结果.进而证明了当n≡0（mod 4）时,图Cn及其r-冠也是交错图.     

辛代数的不变双线性型

考察了辛代数和与它相联系的李三系的双线性型之间的关系,并证明了辛代数的反对称不变双线性型可以唯一扩张到与它相联系的李三系中.作为这种关系的一个应用,得到了二次辛代数是单辛代数的一个充要条件,并证明二次辛代数的唯一分解定理.     

赋值环上的对称双线性型

本讨论赋值环上的对称线性型、二次型和对称矩阵的合同标准形。     

图ω_(4g,4h)的(r_1,r_2,…,r_(4g+4h-1))-冠的优美性

给出了ω_(4g),4h的(r_1,r_2,…,r_(4g)+4h-1)-冠的定义,讨论了ω_(4g),4h的(r_1,r_2,…,r_(4g)+4h-1)-冠的优美性,用构造性的方法给出了图ω_(4g),4h的(r_1,r_2,…,r_(4g)+4h-1)-冠的四种优美标号,并证明了这些ω_(4g),4h的(r_1,r_2,…,r_(4g)+4h-1)-冠也是交错图.     

Graphs of Triangulations and Perfect Matchings

Given a set P of points in general position in the plane, the graph of triangulations of P has a vertex for every triangulation of P, and two of them are adjacent if they differ by a single edge exchange. We prove that the subgraph of , consisting of all triangulations of P that admit a perfect matching, is connected. A main tool in our proof is a result of independent interest, namely that the graph that has as vertices the non-crossing perfect matchings of P and two of them are adjacent if their symmetric difference is a single non-crossing cycle, is also connected.     

3-正则图的不共边的完美匹配（英文）

研究3-正则图的一个有意义的问题是它是否存在k个没有共边的完美匹配.关于这个问题有一个著名的Fan-Raspaud猜想：每一个无割边的3-正则图都有3个没有共边的完美匹配.但这个猜想至今仍未解决.设dim（P（G））表示图G的完美匹配多面体的维数.本文证明了对于无割边的3-正则图G,如果dim（P（G））≤14,那么k≤4：如果dim（P（G））≤20,那么k≤5.     

Reductivity of the Lie Algebra of a Bilinear Form

Let f: V × V → F be a totally arbitrary bilinear form defined on a finite dimensional vector space V over a field F, and let L(f) be the subalgebra of 𝔤𝔩(V) of all skew-adjoint endomorphisms relative to f. Provided F is algebraically closed of characteristic not 2, we determine all f, up to equivalence, such that L(f) is reductive. As a consequence, we find, over an arbitrary field, necessary and sufficient conditions for L(f) to be simple, semisimple or isomorphic to 𝔰𝔩(n) for some n.     

Graphs of Non-Crossing Perfect Matchings

 Let P _n be a set of n=2m points that are the vertices of a convex polygon, and let ℳ _m be the graph having as vertices all the perfect matchings in the point set P _n whose edges are straight line segments and do not cross, and edges joining two perfect matchings M ₁ and M ₂ if M ₂=M ₁−(a,b)−(c,d)+(a,d)+(b,c) for some points a,b,c,d of P _n . We prove the following results about ℳ _m : its diameter is m−1; it is bipartite for every m; the connectivity is equal to m−1; it has no Hamilton path for m odd, m>3; and finally it has a Hamilton cycle for every m even, m≥4. Received: October 10, 2000 Final version received: January 17, 2002 RID="*" ID="*" Partially supported by Proyecto DGES-MEC-PB98-0933 Acknowledgments. We are grateful to the referees for comments that helped to improve the presentation of the paper.     

Smith Normal Form and acyclic matrices

An approach, based on the Smith Normal Form, is introduced to study the spectra of symmetric matrices with a given graph. The approach serves well to explain how the path cover number (resp. diameter of a tree T) is related to the maximal multiplicity MaxMult(T) occurring for an eigenvalue of a symmetric matrix whose graph is T (resp. the minimal number q(T) of distinct eigenvalues over the symmetric matrices whose graphs are T). The approach is also applied to a more general class of connected graphs G, not necessarily trees, in order to establish a lower bound on q(G).     

二分图中含有完美对集的2因子

该文证明若G是2n阶均衡二分图，δ(G)≥(2n-1)/3，则对任何正整数k，n≥4k时，任给G的一个完美对集M，G中存在一个包含M的所有边的恰含k个分支的2 因子(k=1，n=5且δ(G)=3除外）. 特别k=2时，在条件n≥5且δ(G)≥(n+2)/2下，结论也成立. 这里所给的δ(G)的下界是最好的可能.      

Perfect State Transfer on Weighted Abelian Cayley Graphs*

Recently, there are extensive studies on perfect state transfer (PST for short) on graphs due to their significant applications in quantum information processing and quantum computations. However, there is not any general characterization of graphs that have PST in literature. In this paper, the authors present a depiction on weighted abelian Cayley graphs having PST. They give a unified approach to describe the periodicity and the existence of PST on some specific graphs.     

On the Upper Bounds of the Numbers of Perfect Matchings in Graphs with Given Parameters

Let φ(G),κ(G),α(G),χ(G),cl(G),diam(G)denote the number of perfect matchings,connectivity,independence number,chromatic number,clique number and diameter of a graph G,respectively.In this note,by constructing some extremal graphs,the following extremal problems are solved:1.max{φ(G):|V(G)|=2n,κ(G)≤k}=k[(2n-3)!!],2.max{φ(G):|V(G)|=2n,α(G)≥k}=[multiply from i=0 to k-1(2n-k-i)[(2n-2k-1)!!],3.max{φ(G):|V(G)|=2n,χ(G)≤k}=φ(T_(k,2n))T_(k,2n)is the Turán graph,that is a complete k-partite graphon 2n vertices in which all parts are as equal in size as possible,4.max{φ(G):|V(G)|=2n,cl(G)=2}=n1,5.max{φ(G):|V(G)|=2n,diam(G)≥2}=(2n-2)(2n-3)[(2n-5)!!],max{φ(G):|V(G)|=2n,diam(G)≥3}=(n-1)~2[(2n-5)!!].