P5,4m的优美性

设u,v是两个固定顶点,用b条内部互不相交且长度均为a的道路连接u、v所得到的图用Pa,b表示。Kathiresan证实P2r,2m-1(r,m均为任意正整数)是优美的,且猜想:除了(a,b)=(2r 1,4s 2)外,所有的Pa,b都是优美的。杨元生已证实P2r 1,2m-1是优美的,本文证明m=2n(2l-1),(0≤n≤4,l∈N)时,P5,4m是优美图。     

有向图n·■_m的优美性

给定有向图D(V,E),如果存在一个单射f:V(D)→{0,1,…,|E|}使得对于每条有向边(u,v),诱导函数f′:E(D)→{1,2,…,|E|}是一个双射函数,其中,f′(u,v)=[f(v)-f(u)](mod(|E|+1)),则f称为有向图D(V,E)的优美标号,f′称为有向图D(V,E)的诱导的边的优美标号.本文讨论了有向图n.■m的优美性,并且证明了当m=23且n为偶数时,n.■m是优美有向图.     

On the Rigidity of Sparse Random Graphs

A graph with a trivial automorphism group is said to be rigid. Wright proved (Acta Math 126(1) (1971), 1–9) that for a random graph is rigid whp (with high probability). It is not hard to see that this lower bound is sharp and for with positive probability is nontrivial. We show that in the sparser case , it holds whp that G's 2‐core is rigid. We conclude that for all p, a graph in is reconstructible whp. In addition this yields for a canonical labeling algorithm that almost surely runs in polynomial time with o(1) error rate. This extends the range for which such an algorithm is currently known (T. Czajka and G. Pandurangan, J Discrete Algorithms 6(1) (2008), 85–92).     

任意n个完备二分图的并图的k－优美性和算术性

证明了对于正整数k,n,si,ti（si,ti≥2,i=1,2,…,n）,图n/U/i=1,Ksi,ti是k-优美图;对于正整数k,d（d≥2）,k≠0（roodd）及n,si,ti（si,ti≥2,i=1,2,…,n）,图n/U/i=1,Ksi,ti是（k,d）-算术图,前一结论推广了文[6]的相应结果。     

On Distance-Regular Graphs with Height Two

Let be a distance-regular graph with diameter at least three and height h = 2, where . Suppose that for every in and in _d(), the induced subgraph on _d() ₂() is a clique. Then is isomorphic to the Johnson graph J(8, 3).     

On Narrow Hexagonal Graphs with a 3-Homogeneous Suborbit

A connected graph of girth m 3 is called a polygonal graph if it contains a set of m-gons such that every path of length two is contained in a unique element of the set. In this paper we investigate polygonal graphs of girth 6 or more having automorphism groups which are transitive on the vertices and such that the vertex stabilizers are 3-homogeneous on adjacent vertices. We previously showed that the study of such graphs divides naturally into a number of substantial subcases. Here we analyze one of these cases and characterize the k-valent polygonal graphs of girth 6 which have automorphism groups transitive on vertices, which preserve the set of special hexagons, and which have a suborbit of size k – 1 at distance three from a given vertex.     

关于两个有向图n·_3之并的优美性

证实了 ,两个无交有向图 n.C 3之两个相邻 2度点处反方向粘合的优美性 .由于在设计优美标号时 ,缺乏规律性 .从而采用了对顶点数 n,分段设计标号的方法 .     

On Distance-Regular Graphs with Height Two, II

Let be a distance-regular graph with diameter and height , where . Suppose that for every in and every in , the induced subgraph on is isomorphic to a complete multipartite graph with . Then and is isomorphic to the Johnson graph .     

图的割宽极值问题

The problem studied in this paper is to determine e(p, C), the minimum size of a connected graph G with given vertex number p and cut-width C.     

一类优美图

设u、ν是两个固定顶点.用b条内部互不相交且长度皆为a的道路连接u、ν所得的图用P_a,b表示.KM.Kathiresan证实P_2,2m-1(r,m皆为任意正整数)是优美的,且猜想:除了(a,b)=(2r+1,4s+2)外,所有的P_a,b都是优美的.杨元生已证实P_2r+1,2m-1是优美的,并且证实了当r=1,2,3,4时的P_2r,2m也是优美的.本文证实r=5,6,7时P_{2r,2m相似文献}   

渺位六角系统完美匹配Z——变换图

六角系统是2-连通的平面图，其每个内部面都是单位正六边形．六角系统的完美匹配是化学中苯类芳烃体系的Kekule结构．一个六角系统H完美匹配Z—变换图Z（H）是一个图，它的顶点集是H的完匹配集，两个匹配相邻当且仅当它们的对称差是一个单位正六边形．本文用乘积图刻划了沙位六角系统Z—变换图的结构．     

几类有趣图的奇优美性和奇强协调性

对于简单图G=〈V,E〉,如果存在一个映射f:V(G)→{0,1,2,…,2|E|-1}满足:1)对任意的u,v∈V,若u≠v,则f(u)≠f(v);2)max{f(v)|v∈V}=2|E|-1;3)对任意的e_1,e_2∈E,若e_1≠e_2,则g(e_1)≠g(e_2),此处g(e)=|f(u)+f(v)|,e=uv;4)|g(e)|e∈E}={1,3,5,…,2|E|-1},则称G为奇优美图,f称为G的奇优美标号.设G=〈V,E〉是一个无向简单图.如果存在一个映射f:V(G)→{0,1,2,…,2|E|-1},满足:1)f是单射;2)■uv∈E(G),令f(uv)=f(u)+f(v),有{f(uv)|uv∈E(G)}={1,3,5,…,2|E|-1},则称G是奇强协调图,f称为G的.奇强协调标号或奇强协调值.给出了链图、升降梯等几类有趣图的奇优美标号和奇强协调标号.     

Antimagic Labeling of Regular Graphs

A graph is antimagic if there is a one‐to‐one correspondence such that for any two vertices , . It is known that bipartite regular graphs are antimagic and nonbipartite regular graphs of odd degree at least three are antimagic. Whether all nonbipartite regular graphs of even degree are antimagic remained an open problem. In this article, we solve this problem and prove that all even degree regular graphs are antimagic.     

算术图的若干性质

本文得到了算术图的一些新性质，并解决了文[1]提出的两个猜想.     

Zero-divisor Labelings of Graphs

This paper introduces the notions of a zero-divisor labeling and the zero-divisor index of a graph using the zero-divisors of a commutative ring. Viewed in this way, the usual zero-divisor graph is a maximal graph with respect to a zero-divisor labeling. We also study optimal zero-divisor labelings of a finite graph.     

含有多个圈的图的秩

研究了含有多个圈的图的邻接矩阵的秩.将k(k≥2)条点不交的路,首和尾分别粘合得到的图称为Θ-图.用Γ(k-1)表示含有Θ-图作为导出子图的(k-1)-圈图的集合,而用C(η,k)表示含有n个顶点和k个边不交的圈的图的集合.确定了Γ(k-1)中秩等于5和6的图以及C(n,k)中秩等于4,5和6的图.     

《若干并图的优美标号》一文的注

指出了《若干并图的优美标号》一文中的一些不当之处证明了对任意的正整数m和大于1的自然数p,q非连通图(P_2∨(K_m)~(1/2)∪K_(p,q)是优美图.     

Enumerating Planar Locally Finite Cayley Graphs

We characterize the set of planar locally finite Cayley graphs, and give a finite representation of these graphs by a special kind of finite state automata called labeling schemes. As a result, we are able to enumerate and describe all planar locally finite Cayley graphs of a given degree. This analysis allows us to solve the problem of decision of the locally finite planarity for a word-problem-decidable presentation.Mathematics Subject Classiffications (2000). 20F05, 20F10, 20F65, 05C25     

若干并图的优美标号

证明了,对任意大于1的自然数m,n,p,非连通图(■ V ■)∪K_(n,p)是优美图;当k≤p,m=kn+3或m=kn+1时,非连通图(P_2 V ■)∪K_(n,p)是优美图;当p≥2,m=3k+1时,非连通图(P_2 V ■)∪K_(3,p)是优美图;对任意正整数n,p,非连通图(P_1 V P_(2n+2))∪_(n,p)是优美图.     

门槛图与度极大图

证明了门槛图与度极大图是一类图的两种不同说法,同时用图的对角限制极左矩阵刻画这一类图的结构。