关于Kn^-t的点可区别正常边染色

一个图的边染色称为是点可区别的，如果任意两个不同的顶点的关联边的颜色的集合不同．设Kn^-t表示从n阶完全图中删去t条彼此不相邻的边后所得到的图．本文对Kn^-t的点可区别正常边染色进行了讨论．     

拟正则完全二部图的局部最可靠性

用P(G，ρ)表示顶点完全可靠，而边则以概率ρ∈(0，1)相互独立地出现故障的图G的全终端不可靠度，即G因边故障而变得不连通的概率．本文证明了边故障率ρ充分小时，拟正则完全二部图在具有相同点数和相同边数的图类中是惟一具有最小全终端不可靠度的图．     

关于K-tn的点可区别正常边染色

一个图的边染色称为是点可区别的,如果任意两个不同的顶点的关联边的颜色的集合不同. 设K-tn表示从n阶完全图中删去t条彼此不相邻的边后所得到的图. 本文对K-tn的点可区别正常边染色进行了讨论.     

关于(K_n~-)~t 的点可区别正常边染色(英文)

一个图的边染色称为是点可区别的 ,如果任意两个不同的顶点的关联边的颜色的集合不同 .设K-tn 表示从 n阶完全图中删去 t条彼此不相邻的边后所得到的图 .本文对 K-tn 的点可区别正常边染色进行了讨论 .     

随机偏好连接图的中心极限定理

我们研究了一类具有随机顶点和边的随机连接图模型, 其中顶点的随机性由一个Poisson 点过程所决定, 边的随机性由一个概率连接函数所决定. 我们得到了带偏好的随机连接图模型的关于所有随机边的长度和的一个中心极限定理.     

一类偏向删点及顶点有限制的随机图上的相变

固定α_0∈[0,1)及β∈[0,1/2).该文引入如下随机图过程(G_t)t≥1:设在时刻1及2已存在图G_1=G_2,其中G_1的顶点为v_1,v_2且它们之间有2条边相连.当t≥3时,G_t定义如下:(i)G_(t-1)中任意顶点v不活跃的概率为α_0.顶点不活跃意味着其不能与t时刻新增加的顶点相连.此概率独立于自己以及其他顶点t-1之前的状态;(ii)以概率1-β增加一个新顶点v_t.在G_(t-1)中以概率dw(t-1)/∑vdv(t-1)选一顶点w,其中d_w(t-1)表w在G_(t-1)中的度.若w是活跃的则在v_t与w之间连1条边,否则在v_t上加个环;(iii)以概率β在G_(t-1)中删去一顶点u,其中u被选中的概率为(1-du(t-1)/∑vdv(t-1))/(n_(t-1)-1).此处,n_(t-1)是G_(t-1)的顶点个数.令N_k(t)表G_t中度为k的顶点个数.该文证明了G_t度分布的期望在2β/1-α_0=1附近存在一相变:当2β/1-α_01时,N_k(t)/t的期望是呈指数衰减的;当2β/1-α_01时,N_k(t)/t的期望是呈幂律衰减的.     

K_(n,n)的局部最优可靠性

假设n点m边的简单无向图G=(V,E)的每个顶点完全可靠,各边相互独立地以同一概率q(0q1)发生故障,则用G不连通的概率P(G,q)作为衡量网不可靠程度的指标.如果对于充分接近q0的所有q都有P(G,q)P(H,q),则称在边故障概率q～q0时,网络G比H可靠.证明了当q～0时,Kn,n(n4)是2n点n2边图中局部最优可靠的.     

K1,p-受限图

图G中同构于Ki,p的子图叫G的p-爪(P≥3)．如果G中任意一个p-爪中1度顶点之间边(在G中的边)的数目≥P-2,则称G为K1,p-受限图,它是无爪图的推广．本文证明了连通、局部2-连通的K1,4-受限图是完全圈可扩的．     

NIC-平面图的存活率

假设火在图G的某个顶点燃起,消防员每步最多可以防护k个顶点,然后火蔓延到所有未被防护的邻点.当火随机地在图G的一个顶点燃起时,消防员最多能防护的顶点数的平均比率称为图G的k-存活率,记为ρ_k(G).如果图G能画在平面上使得每两对交叉边至多有一个公共顶点,那么称G是NIC-平面图.本文证明了NIC-平面图G有ρ₅(G)> 1/73.     

关于哈密顿线图的一个注记

一、 引言令 G 是顶点集合为 V(G)且边集合为 E(G)的简单图.图 G 的线图 L(G)是顶点集合为 E(G)的图,L(G)的两个顶点,e_1和 e_2是相邻接的当且仅当 e_1和 e_2在 G中有一个公共顶点.图 G 的一条通道是点与边的一个交替序列 v_0,e_1,v_1,…,v_(n-1),e_n,v_n 其中 e_i(i=     

Graph evolution by stochastic additions of points and lines

We define the graph G₁ recursively from G_t?1 by adding a point or a line with probability ? and q, respectively, if G_t?1 is not complete; if G_t?1 is complete we always add a point. By using recursions we investigate the probability distribution of the order and size of G_? of the minimum and maximum orders for a fixed size, and of the minimum and maximum sizes for a fixed order. Expected values and generating functions are determined.     

On bipartite zero-divisor graphs

A (finite or infinite) complete bipartite graph together with some end vertices all adjacent to a common vertex is called a complete bipartite graph with a horn. For any bipartite graph G, we show that G is the graph of a commutative semigroup with 0 if and only if it is one of the following graphs: star graph, two-star graph, complete bipartite graph, complete bipartite graph with a horn. We also prove that a zero-divisor graph is bipartite if and only if it contains no triangles. In addition, we give all corresponding zero-divisor semigroups of a class of complete bipartite graphs with a horn and determine which complete r-partite graphs with a horn have a corresponding semigroup for r≥3.     

给定顶点数和边数的连通图的Q-谱半径的界

图的无符号拉普拉斯矩阵是图的邻接矩阵和度对角矩阵的和,其特征值记为q1≥q2≥…≥qn.设C(n,m)是由n个顶点m条边的连通图构成的集合,这里1≤n-1≤m≤(n2).如果对于任意的G∈C(n,m)都有q1(G*)≥q1(G)成立,图G*∈C(n,m)叫做最大图.这篇文章证明了对任意给定的正整数a=m-n+1,如果n...     

The Zero-Divisor Graphs Which Are Uniquely Determined By Neighborhoods

A nonempty simple connected graph G is called a uniquely determined graph, if distinct vertices of G have distinct neighborhoods. We prove that if R is a commutative ring, then Γ(R) is uniquely determined if and only if either R is a Boolean ring or T(R) is a local ring with x² = 0 for any x ∈ Z(R), where T(R) is the total quotient ring of R. We determine all the corresponding rings with characteristic p for any finite complete graph, and in particular, give all the corresponding rings of K_n if n + 1 = p^q for some primes p, q. Finally, we show that a graph G with more than two vertices has a unique corresponding zero-divisor semigroup if G is a zero-divisor graph of some Boolean ring.     

3-连通3-正则图中的可序子集

图G的k元点集X={x1,x2,…,xk}被称为G的k-可序子集,如果X的任意排列都按序排在G的某个圈上.称G是k-可序图,如果G的每一个k元子集都是G的k-可序子集.称G为k-可序Hamilton图,如果X的任意排列都位于G的Hamilton圈上.研究了3-连通3-正则图的可序子集的存在性问题.     

The random planar graph process

We consider the following variant of the classical random graph process introduced by Erd?s and Rényi. Starting with an empty graph on n vertices, choose the next edge uniformly at random among all edges not yet considered, but only insert it if the graph remains planar. We show that for all ε > 0, with high probability, θ(n²) edges have to be tested before the number of edges in the graph reaches (1 + ε)n. At this point, the graph is connected with high probability and contains a linear number of induced copies of any fixed connected planar graph, the first property being in contrast and the second one in accordance with the uniform random planar graph model. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

OD-CHARACTERIZATION OF ALMOST SIMPLE GROUPS RELATED TO U6（2）

Let G be a finite group and π(G) = { p 1 , p 2 , ··· , p k } be the set of the primes dividing the order of G. We define its prime graph Γ(G) as follows. The vertex set of this graph is π(G), and two distinct vertices p, q are joined by an edge if and only if pq ∈π e (G). In this case, we write p ～ q. For p ∈π(G), put deg(p) := |{ q ∈π(G) | p ～ q }| , which is called the degree of p. We also define D(G) := (deg(p 1 ), deg(p 2 ), ··· , deg(p k )), where p 1 < p 2 < ··· < p k , which is called the degree pattern of G. We say a group G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and degree pattern as G. Specially, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Let L := U 6 (2). In this article, we classify all finite groups with the same order and degree pattern as an almost simple groups related to L. In fact, we prove that L and L.2 are OD-characterizable, L.3 is 3-fold OD-characterizable, and L.S 3 is 5-fold OD-characterizable.     

Kp,q-factorization of complete bipartite graphs

Let K_(m,n) be a complete bipartite graph with two partite sets having m and nvertices, respectively. A K_(p,q)-factorization of K_(m,n) is a set of edge-disjoint K_(p,q)-factorsof K_(m,n) which partition the set of edges of K_(m,n). When p=i and q is a prime number,Wang, in his paper "On K_(1,k)-factorizations of a complete bipartite graph" (Discrete Math,1994, 126; 359-364), investigated the K_(1,q)-factorization of K_(m,n) and gave a sufficientcondition for such a factorization to exist. In the paper "K_(1,k)-factorizations of completebipartite graphs" (Discrete Math, 2002, 259: 301-306), Du and Wang extended Wang'sresult to the case that q is any positive integer In this paper, we give a sufficient conditionfor K_(m,n) to have a K_(p,q)-factorization. As a special case, it is shown that the Martin's BACconjecture is true when p: q=k: (k+1) for any positive integer k.     

几类多圈图的拉普拉斯谱刻画

设图G是一个简单连通图. 如果任何一个与图G同拉普拉斯谱的图都与图G同构,则称图G是由其拉普拉斯谱确定的. 定义了双圈图\theta_{n}(p_1,p_2,\cdots,p_t) 和m 圈图H_n(m\cdot C_3;p_1,p_2,\cdots,p_t). 证明了双圈图\theta_{n}(p)和\theta_{n}(p,q),三圈图H_n(3\cdot C_3;p)和H_n(3\cdot C_3;p,q)分别是由它们的拉普拉斯谱确定的.     

INVERSE MONOIDS OF GRAPHS

. IntroductionGraph endomorphism and its regularity property have been investigated in some literatures (of. [1--41 for examples). The invertibility is a stronger algebraic property thanregUlarity in semigroup theory. It is commonly agreed that inverse semigroups are the mostpromising class of semigroups for study. In this paper we first present a combinatorial characterization of an inverse monoid of a graph (Theorem 2.3). Then using this we prove thata bipartite graph with an inverse monoi…