完全图字典乘积的强半径和强直径

目前针对乘积图的强距离研究已经取得丰富成果,主要给出完全图字典乘积的强距离结果.基于字典乘积图与其因子图的关系,确定了两个完全图字典乘积的最小强半径;利用因子图与字典乘积图的顶点数的关系,得到完全图字典乘积的最大强直径和最大强半径的范围.除此之外,通过字典乘积的结合性,将字典乘积图的相关强距离结果进一步推广到多个完全图的字典乘积.     

k-强传递阵

给出k-强传递阵的几个性质，讨论k-强传递阵与强传递阵的关系，并且指出强传递与准传递是等价的。     

图的距离谱综述

设D(G)为连通图G的距离矩阵,λ1(D)≥>…≥AnD)是D(G)的特征值.距离特征值的研究可追溯到Graham 和Pollack [Bell Syst.Tech.J.,1971,50:2495-2519]的工作,其中描述了负距离特征值数目与数据通信系统寻址问题之间的关系.2014年,Aouchiche和Hansen...     

含有两个非临界点的强连通定向图的弧数

证明顶点数为$n\geq 4$,弧数为$m\geq {n-1 \choose 2}+3$的强连通定向
图$D$中存在两点$u^*$、!$v^*$,使得$D-u^*$和$D-v^*$都是强连通的, 并用例子说明这里所给的
关于弧数的下界是紧的.     

链状正则图的平均距离

本文构造了一类链状正则图G_k∶δ,求出了它们的平均距离D(G_k.δ),并得到关系式上式等号成立当且仅当δ=4f且k=0.这个估计式指出了施容华猜想[1]D(G)≤n/(δ 1)不成立. 文中进一步证明了这一类链状正则图有最大的直径,所以可以作出猜想: 若G是n阶连通图,则D(G)<(n 1)/(δ 1),其中δ是图G的最小度。     

n阶本原极小强连通有向图k-指数的最小值

本文证明了:当1≤k≤︱n/4︱时,n阶本原极小强连通有向图k指数的最小值是4。     

距离正则图的推广

本文研究了直径为d(Γ) ≥ 2的距离正则图Γ的补图.利用Γ的交叉数分别证明了当d=2时,Γ的补图式强正则;当d ≥ 3时,Γ的补图是广义强正则.将文献[2]中的距离正则图Grassmann图、对偶极图、Hamming图推广到它们的补图,从而得到广义强正则图.     

k-边连通图的新充分谱条件

令G是一个简单连通图.如果连通图G被删除少于k条边后仍然保持连通,则称G是k-边连通的.基于图G或补图■的距离谱半径,距离无符号拉普拉斯谱半径,Wiener指数和Harary指数,提供了图G是k-边连通的新充分谱条件,从而建立了图的代数性质与结构性质之间的紧密联系.     

关于图的直径和平均距离

图的直径和平均距离是度量网络有效性的两个重要参数．Ore通过图的顶点数和直径给出无向图的最大边数．Entringer，Jakson，Slater和Ng，Teh通过图的顶点数和边数分别给出无向图和有向图平均距离的下界．该文提供这两个结果的简单证明，给出有向图类似Ore的结果，并通过图的直径改进Entringer等人的结果到更一般的情形．结合本文和Ore的结果，可以得到一个无向图和有向图平均距离的下界，它比Plesnik得到的下界更好．     

五角链距离谱半径的极图

连通图G的距离谱半径是其距离矩阵的最大特征值.为了刻画五角链距离谱半径达到最大值和最小值时的极图结构,通过引入图的变换,结合代数图论相关知识,找到了五角链距离谱的变化规律,从而得出在所有含有n个正五边形的五角链中,距离谱半径最小的极图为第一类五角链L_n,距离谱半径最大的极图为第二类五角链T_n.     

双圈图的距离无符号拉普拉斯谱半径

连通图$G$的距离无符号拉普拉斯矩阵定义为$\mathcal{Q}(G)=Tr(G)+D(G)$, 其中$Tr(G)$和$D(G)$分别为连通图$G$的点传输矩阵和距离矩阵. 图$G$的距离无符号拉普拉斯矩阵的最大特征值称为$G$的距离无符号拉普拉斯谱半径. 本文确定了给定点数的双圈图中具有最大的距离无符号拉普拉斯谱半径的图.     

一类联图的距离谱半径以及盖理论（英文）

令X=(n₁,n₂,…,n_t),Y=(m₁,m₂,…,m_t)是两个t维递减序列.如果对所有的j,1≤j≤t,都有∑_i=1~j、n_i≥∑_i=1~j m_i以及∑_i=1~t n_i=∑_i=1~t m_i,则称X可盖Y,记作X■Y.如果X≠Y,则记作X■Y.本文考虑联图G(n₁,n₂,…,n_t;a)=(K_n1∪_n2∪…∪K_nt)∨K_a的谱半径,这里n₁+n₂+…+n_t+a=n,(n₁,n     

On the Mean Distance in Scale Free Graphs

We consider a graph, where the nodes have a pre-described degree distribution F, and where nodes are randomly connected in accordance to their degree. Based on a recent result (R. van der Hofstad, G. Hooghiemstra and P. Van Mieghem, “Random graphs with finite variance degrees,” Random Structures and Algorithms, vol. 17(5) pp. 76–105, 2005), we improve the approximation of the mean distance between two randomly chosen nodes given by M. E. J. Newman, S. H. Strogatz, and D. J. Watts, “Random graphs with arbitrary degree distribution and their application,” Physical Review. E vol. 64, 026118, pp. 1–17, 2001. Our new expression for the mean distance involves the expectation of the logarithm of the limit of a super-critical branching process. We compare simulations of the mean distance with the results of Newman et al. and with our new approach. AMS 2000 Subject Classification: 05C80, 60F05     

图的距离无符号拉普拉斯谱半径的关于色数的下界

Let G be a connected graph on n vertices with chromatic number k, and let ρ(G)be the distance signless Laplacian spectral radius of G. We show that ρ(G) ≥ 2n + 2「n k」- 4,with equality if and only if G is a regular Tur′an graph.     

Lower and upper orientable strong radius and strong diameter of complete k-partite graphs

For two vertices u and v in a strong digraph D, the strong distance sd(u,v) between u and v is the minimum size (the number of arcs) of a strong sub-digraph of D containing u and v. For a vertex v of D, the strong eccentricity se(v) is the strong distance between v and a vertex farthest from v. The strong radius srad(D) (resp. strong diameter sdiam(D)) is the minimum (resp. maximum) strong eccentricity among the vertices of D. The lower (resp. upper) orientable strong radius srad(G) (resp. SRAD(G)) of a graph G is the minimum (resp. maximum) strong radius over all strong orientations of G. The lower (resp. upper) orientable strong diameter sdiam(G) (resp. SDIAM(G)) of a graph G is the minimum (resp. maximum) strong diameter over all strong orientations of G. In this paper, we determine the lower orientable strong radius and diameter of complete k-partite graphs, and give the upper orientable strong diameter and the bounds on the upper orientable strong radius of complete k-partite graphs. We also find an error about the lower orientable strong diameter of complete bipartite graph K_m,n given in [Y.-L. Lai, F.-H. Chiang, C.-H. Lin, T.-C. Yu, Strong distance of complete bipartite graphs, The 19th Workshop on Combinatorial Mathematics and Computation Theory, 2002, pp. 12-16], and give a rigorous proof of a revised conclusion about sdiam(K_m,n).     

图的无符号拉普拉斯和距离无符号拉普拉斯特征值

Let G be a simple graph. We first show that ■, where δiand di denote the i-th signless Laplacian eigenvalue and the i-th degree of vertex in G, respectively.Suppose G is a simple and connected graph, then some inequalities on the distance signless Laplacian eigenvalues are obtained by deleting some vertices and some edges from G. In addition, for the distance signless Laplacian spectral radius ρQ(G), we determine the extremal graphs with the minimum ρQ(G) among the trees with given diameter, the unicyclic and bicyclic graphs with given girth, respectively.     

On d-arc-dominated Oriented Graphs

We consider only digraphs that are oriented graphs, meaning orientations of simple finite graphs. An oriented graph D = (V, A) with minimum outdegree d is called d-arc-dominated if for every arc \({(x, y) \in A}\) there is a vertex \({u \in V}\) with outdegree d such that both \({(u, x) \in A}\) and \({(u, y) \in A}\) hold. In this paper, we show that for any integer d ≥ 3 the girth of a d-arc-dominated oriented graph is less than or equal to d. Moreover, for every integer t with 3 ≤ t ≤ d there is a d-arc-dominated oriented graph with girth t. We also give a characterization for oriented graphs with both minimum outdegree and girth d to be d-arc-dominated and classify all d-arc-dominated d-circular oriented graphs with girth d.     

强正则图的一些性质

文[3]给出了强正则图的概念及有关性质,本文在此基地上利用图的谱性质,得到了强正则图的又一些性质。