Two‐sided Group Digraphs and Graphs

We study a family of digraphs (directed graphs) that generalises the class of Cayley digraphs. For nonempty subsets of a group G, we define the two‐sided group digraph to have vertex set G, and an arc from x to y if and only if for some and . In common with Cayley graphs and digraphs, two‐sided group digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on L and R under which may be viewed as a simple graph of valency , and we call such graphs two‐sided group graphs. We also give sufficient conditions for two‐sided group digraphs to be connected, vertex‐transitive, or Cayley graphs. Several open problems are posed. Many examples are given, including one on 12 vertices with connected components of sizes 4 and 8.     

图的k部划分的一个注记

设k≥2是一个整数。本文证明了任意有m条边的图都存在一个顶点的划分V_1,V_2…,V_k,使得e(V_1,V_2…,V_k)≥k-1/k m+k-1/2k((2m+1/4)~1/2-1/2)-(k-2)~2/8k,且max{e(V_i):1≤i≤k}≤m/k~2+(k-1)/2k~2((2m+1/4)~1/2-1/2+3/8-7k-4/8k~2.我们的结果改进了[Fan G.,Hou J.,Zeng Q.,A bound for judicious k-partitions of graphs,Discrete Appl.Math.,2014,179:86—99]的主要结论.     

Continuous-Time Independent Edge-Markovian Random Graph Process

In this paper, the continuous-time independent edge-Markovian random graph process model is constructed. The authors also define the interval isolated nodes of the random graph process, study the distribution sequence of the number of isolated nodes and the probability of having no isolated nodes when the initial distribution of the random graph process is stationary distribution, derive the lower limit of the probability in which two arbitrary nodes are connected and the random graph is also connected, and prove that the random graph is almost everywhere connected when the number of nodes is sufficiently large.     

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.     

Distance Integral Complete Multipartite Graphs with s=5, 6

Let D(G) = (dij )n×n denote the distance matrix of a connected graph G with order n, where dij is equal to the distance between vertices vi and vj in G. A graph is called distance integral if all eigenvalues of its distance matrix are integers. In 2014, Yang and Wang gave a su?cient and necessary condition for complete r-partite graphs Kp1,p2,··· ,pr =Ka1·p1,a2·p2,··· ,as···ps to be distance integral and obtained such distance integral graphs with s = 1, 2, 3, 4. However distance integral complete multipartite graphs Ka1·p1,a2·p2,··· ,as·ps with s>4 have not been found. In this paper, we find and construct some infinite classes of these distance integral graphs Ka1·p1,a2·p2,··· ,as·ps with s = 5, 6. The problem of the existence of such distance integral graphs Ka1·p1,a2·p2,··· ,as·ps with arbitrarily large number s remains open.     

Spanning embeddings of arrangeable graphs with sublinear bandwidth

The Bandwidth Theorem of Böttcher, et al. [Mathematische Annalen 343 (2009), 175–205] gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded maximum degree. We generalise this result to a‐arrangeable graphs H with , where n is the number of vertices of H. Our result implies that sufficiently large n‐vertex graphs G with minimum degree at least contain almost all planar graphs on n vertices as subgraphs. Using techniques developed by Allen, et al. [Combinatorica 33 (2013), 125–160] we can also apply our methods to show that almost all planar graphs H have Ramsey number at most . We obtain corresponding results for graphs embeddable on different orientable surfaces. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 270–289, 2016     

Random walks on stochastic hyperbolic half planar triangulations

We study the simple random walk on stochastic hyperbolic half planar triangulations constructed in (Angel and Ray, Ann Probab, in press). We show that almost surely the walker escapes the boundary of the map in positive speed and that the return probability to the starting point after n steps scales like © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 213–234, 2016     

新型多孔芳香骨架材料用于硝基爆炸物的检测

采用付玫瑰苯胺与对苯二甲醛通过希夫碱偶联反应制备了新型的多孔芳香骨架材料.经过FTIR,TGA,PXRD,SEM,TEM和Ar吸附等分析方法对多孔芳香骨架材料的成键方式和骨架结构进行了表征.分析结果显示,该多孔芳香骨架材料具有优异的热稳定性(350℃仅失重5%)和溶剂稳定性,其Langmuir比表面积约为472m~2/g.紫外光谱和荧光光谱测试分析表明,该材料在苯、甲苯和氯苯等芳香化合物环境中无明显的荧光强度变化.然而在硝基爆炸物环境中会发生灵敏的、专一的荧光淬灭现象.该多孔材料可应用于硝基爆炸物的检测.     

Eigenvalues of the k‐th power of a graph

The k‐th power of a graph G, denoted by , is a graph with the same set of vertices as G such that two vertices are adjacent in if and only if their distance in G is at most k. In this paper, we give the bounds on the spectral radius of and . The Nordhaus–Gaddum‐type inequality for the spectral radius of the graph is also presented. Moreover, we obtain an upper bound on the energy of the second power of graphs.     

A unique continuation result for the plate equation and an application

In this paper, we prove the unique continuation property for the weak solution of the plate equation with the low regular coefficient. Then, we apply this result to study the global attractor for the semilinear plate equation with a localized damping. Copyright © 2015 John Wiley & Sons, Ltd.