张量乘积图的同构因子分解

本文讨论了两个可分图的张量乘积图的同构因子分解问题.给出了张量乘积图可同构因子分解的判定条件.     

循环图的Adam同构的性质

本文得到了任意两个连通循环图是(?)d(?)m同构的充要条件,并且还得到两个连通循环图是(?)d(?)m同构的另一必要条件。     

图模式挖掘中的子图同构算法

图模式挖掘问题在Web挖掘、生物信息学、社会关系等众多领域有广泛的应用,它涉及到子图的搜索以及子图的同构问题.这两个问题都具有相当高的计算复杂度,现有的子图同构问题大多采用最小编码算法,但对无标签图特别是对无标签无向图,该算法效率较底,从而子图的同构成为图模式挖掘问题的一个瓶颈.针对无标签图,以代数理论为基础,分别利用度序列和特征值构造了两种子图同构算法,用于对有向图和无向图的同构判别.最后对2个真实生物网络进行了仿真实验,结果表明,算法的效率优于现有算法.     

黑白顶点数目相等的无完美匹配四角系统

四角系统是一个二部图，二部图有完美匹配的一个必要条件是对其顶点进行正常着色后，两个色类所含的顶点数相等，然而这一条件并不充分，本文利用构造法证明了两个色类所含顶点数相等却无完美匹配的四角系统的最小阶数是14，并且只有3种非同构的形状，由本文的方法还可以进一步构造出15阶和16阶无完美匹配四角系统的所有非同构形状，它们的数目分别是22与155。     

I-hedrite图平面嵌入的唯一性

本文研究每一个面圈的圈长仅为2,3或4的无割点的4·正则连通平面图,称之为I-hedrite图.证明在相等意义上,I-hedrite图的平面嵌入是唯一的.这个唯一性结论意味着,两个i-hedrite图(即每一个面的度仅为2,3或4的4-正则连通平图)是相等的当且仅当它们是同构的,从而解决了i-hedrite图的同构构造在相等意义上的唯一性问题.     

关于四元数代数同构的条件

引入广义四元数代数的 K上表示矩阵的概念 ,探讨复线性表示与 K上表示矩阵的关系 .在此基础上 ,给出两个广义四元数代数同构的一个新的判定条件     

几类图的匹配等价图类

两个图G和H的匹配多项式相等,则称它们匹配等价.用[G]表示图G的所有不同构的匹配等价图的集合.刻画了匹配次大根小于1的图及这些图的补图的匹配等价图类.     

论双变量同构凸函数

在定义加权同构平均值的基础上将凸函数、几何凸函数、平方凸函数等理论统一为双变量同构凸函数理论,给出其统一的微分判别法则,并由此推导出不同类型的同构平均值间不等关系的简单判定法则.     

完全二部图的Kp,p-分解大集的存在谱北大核心CSCD

令H,G是两个简单图,G是H的一个子图.H的G-分解,记为（λH,G）-GD,是指将图λH的所有边分拆为若干个与G同构的子图（称为G-区组）.H的G-分解的大集,记为（λH,G）-LGD,是指图H的所有与G同构的子图的一个分拆Β1,Β2,…,Βm,使得每个Bj（1≤j≤m）为一个（λH,G）-GD （称为小集）.本文中,我们对完全二部图的K（p,p）-分解的大集进行了研究,利用Kv的λ重Kκ-因子大集的存在性结果,采用直接构造的方法,得到了大集（λK（m,n）,K（p,p））-LGD的存在谱,其中p为任意素数.     

完全二部图的K_(p,p)-分解大集的存在谱

令H,G是两个简单图,G是H的一个子图.H的G-分解,记为(λH,G)-GD,是指将图λH的所有边分拆为若干个与G同构的子图(称为G-区组).H的G-分解的大集,记为(λH,G)-LGD,是指图H的所有与G同构的子图的一个分拆Β_1,Β_2,…,Β_m,使得每个B_j(1≤j≤m)为一个(λH,G)-GD (称为小集).本文中,我们对完全二部图的K_(p,p)-分解的大集进行了研究,利用K_v的λ重K_κ-因子大集的存在性结果,采用直接构造的方法,得到了大集(λK_(m,n),K_(p,p))-LGD的存在谱,其中p为任意素数.     

The power graph of a finite group

The power graph of a group is the graph whose vertex set is the group, two elements being adjacent if one is a power of the other. We observe that non-isomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs must be isomorphic. We conjecture that two finite groups with isomorphic power graphs have the same number of elements of each order. We also show that the only finite group whose automorphism group is the same as that of its power graph is the Klein group of order 4.     

Locally homogeneous graphs from groups

A graph is called locally homogeneous if the subgraphs induced at any two points are isomorphic. in this Note we give a method for constructing locally homogeneous graphs from groups. the graphs constructable in this way are exactly the locally homogeneous graphs with a point symmetric universal cover. As an example we characterize the graphs that are locally n-cycles.     

Note on isomorphic hypergraphs and some extensions of Whitney's theorem to families of sets

H. Whitney proved that, apart from a simple exeptional case, whenever the line graphs of two finite graphs are isomorphic then so are the graphs themselves. In this note (i) similar results are proved for finite hypergraphs, (ii) it is shown that certain extensions of Whitney's theorem to hypergraphs are false, (iii) a Whitney-type theorem is established for infinite hypergraphs.     

Isomorphisms of finite semi-Cayley graphs

Let G be a finite group. A Cayley graph over G is a simple graph whose automorphism group has a regular subgroup isomorphic to G. A Cayley graph is called a CI-graph(Cayley isomorphism) if its isomorphic images are induced by automorphisms of G. A well-known result of Babai states that a Cayley graph Γ of G is a CI-graph if and only if all regular subgroups of Aut(Γ) isomorphic to G are conjugate in Aut(Γ). A semi-Cayley graph(also called bi-Cayley graph by some authors) over G is a simple graph whose automorphism group has a semiregular subgroup isomorphic to G with two orbits(of equal size). In this paper, we introduce the concept of SCI-graph(semi-Cayley isomorphism)and prove a Babai type theorem for semi-Cayley graphs. We prove that every semi-Cayley graph of a finite group G is an SCI-graph if and only if G is cyclic of order 3. Also, we study the isomorphism problem of a special class of semi-Cayley graphs.     

Indistinguishable Trees and Graphs

We show that a number of graph invariants are, even combined, insufficient to distinguish between non-isomorphic trees or general graphs. Among these are: the spectrum of eigenvalues (equivalently, the characteristic polynomial), the number of independent sets of all sizes or the number of connected subgraphs of all sizes. We therefore extend the classical theorem of Schwenk that almost every tree has a cospectral mate, and we provide an answer to a question of Jamison on average subtree orders of trees. The simple construction that we apply for this purpose is based on finding graphs with two distinguished vertices (called pseudosimilar) that do not belong to the same orbit but whose removal yields isomorphic graphs.     

P3-isomorphisms for graphs

The P₃-graph of a finite simple graph G is the graph whose vertices are the 3-vertex paths of G, with adjacency between two such paths whenever their union is a 4-vertex path or a 3-cycle. In this paper we show that connected fnite simple graphs G and H with isomorphic P₃-graphs are either isomorphic or part of three exceptional families. We also characterize all isomorphisms between P₃-graphs in terms of the original graphs. © 1997 John Wiley & Sons, Inc. J Graph Theory 26:35–51, 1997     

基于拉普拉斯谱确定的两类树

设图G是简单连通图.如果任何一个与图G关于拉普拉斯矩阵同谱的图,都与图G同构,称图G可由其拉普拉斯谱确定.定义了树Y_n和树F(2,n,1)两类特殊结构的树.利用同谱图线图的特点,证明了树Y_n和树F(2,n,1)可由其拉普拉斯谱确定.     

A natural graph of finite fields distinguishing between models

We define a graph structure associated in a natural way to finite fields that nevertheless distinguishes between different models of isomorphic fields. Certain basic notions in finite field theory have interpretations in terms of standard graph properties. We show that the graphs are connected and provide an estimate of their diameter. An accidental graph isomorphism is uncovered and proved. The smallest non-trivial Laplace eigenvalue is given some attention, in particular for a specific family of 8-regular graphs showing that it is not an expander. We introduce a regular covering graph and show that it is connected if and only if the root is primitive.     

Some results on the reconstruction problems. p-claw-free,chordal, and p4-reducible graphs

A claw is an induced subgraph isomorphic to K_1,3. The claw-point is the point of degree 3 in a claw. A graph is called p-claw-free when no p-cycle has a claw-point on it. It is proved that for p ≥ 4, p-claw-free graphs containting at least one chordless p-cycle are edge reconstructible. It is also proved that chordal graphs are edge reconstructible. These two results together imply the edge reconstructibility of claw-free graphs. A simple proof of vertex reconstructibility of P₄-reducible graphs is also presented. © 1995 John Wiley & Sons, Inc.     

Which finitely generated Abelian groups admit isomorphic Cayley graphs?

We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.