一类半对称图的构造

称一个有限简单无向图X是半对称图,如果图X是正则的且边传递但非点传递.主要利用仿射几何构造了一类2p~n阶连通p~3。度的半对称图的无限族,其中p≥n≥8.     

Vertex-distinguishing E-total Coloring of Complete Bipartite Graph K_(7,n) when 7≤n≤95

Let G be a simple graph. A total coloring f of G is called an E-total coloring if no two adjacent vertices of G receive the same color, and no edge of G receives the same color as one of its endpoints. For an E-total coloring f of a graph G and any vertex x of G, let C(x) denote the set of colors of vertex x and of the edges incident with x, we call C(x) the color set of x. If C(u)≠ C(v) for any two different vertices u and v of V(G), then we say that f is a vertex-distinguishing E-total coloring of G or a VDET coloring of G for short. The minimum number of colors required for a VDET coloring of G is denoted by χ_(vt)~e(G) and is called the VDET chromatic number of G. The VDET coloring of complete bipartite graph K_(7,n)(7 ≤ n ≤ 95) is discussed in this paper and the VDET chromatic number of K_(7,n)(7 ≤ n ≤ 95) has been obtained.     

出生率与年龄段有关的生灭分枝树

本文基于生灭过程的生灭演化机理, 将生物繁衍过程描述为有向随机图过程-随机分枝树, 建立了出生率与年龄段有关的生灭分枝树演化模型. 本文研究了任一节点在不同年龄及临死时刻的出度分布、虚出度分布和拟出度分布, 并证明了拟出度过程是随机时刻终止的Poisson过程, 讨论了首生年龄及相对出生年龄的分布, 给出了任一节点成为孤立节点的概率.     

On Symmetric Cayley Graphs of Valency Eleven

A Cayley graph Γ=Cay(G,S)is said to be normal if G is normal in Aut Γ.In this paper,we investigate the normality problem of the connected 11-valent symmetric Cayley graphs Γ of finite nonabelian simple groups G,where the vertex stabilizer Av is soluble for A=Aut Γ and v ∈ VΓ.We prove that either Γ is normal or G=A5,A10,A54,A274,A549 or A1099.Further,11-valent symmetric nonnormal Cayley graphs of A5,A54 and A274 are constructed.This provides some more examples of nonnormal 11-valent symmetric Cayley graphs of finite nonabelian simple groups after the first graph of this kind(of valency 11)was constructed by Fang,Ma and Wang in 2011.     

构造纠缠目击的一般方法

量子纠缠作为量子通信和量子计算过程中不可缺少的资源,在量子信息领域中有着广泛的应用.如何判定一个给定的量子态是否为纠缠态仍然是一个重要的课题.纠缠目击是一种特殊的自伴算子,它可以用来判断一个量子态是否为纠缠态.本文首先从纠缠目击的定义入手,给出构造纠缠目击的一般方法,证明了当一个可测量A在可分纯态上的最大期望CA严格小于它的最大特征值λ_(max)(A)时,对任何满足条件C_A≤Cλ_(max)(A)的参数C,算子W_C=CI-A都是一个纠缠目击;然后,作为应用得到了利用图态的稳定子构造纠缠目击的一系列方法.     

具有五个不同于0和1的Laplacian特征值的单圈图

设$U$是$n$阶单圈图, $m_{U}(1)$是$U$的拉普拉斯特征值1的重数.众所周知,0是连通图重数为1的拉普拉斯特征值.这意味着如果$U$有五个不同于0和1的拉普拉斯特征值,那么$m_U(1)=n-6$.本文完整刻画了$m_U(1)=n-6$的所有单圈图.     

An Improved Upper Bound on the Linear 2-arboricity of 1-planar Graphs

The linear 2-arboricity la_2(G) of a graph G is the least integer k such that G can be partitioned into k edge-disjoint forests,whose component trees are paths of length at most 2.In this paper,we prove that if G is a 1-planar graph with maximum degree Δ,then la_2(G)≤[(Δ+1)/2]+7.This improves a known result of Liu et al.(2019) that every 1-planar graph G has la_2(G)≤[(Δ+1)/2]+14.We also observe that there exists a 7-regular 1-planar graph G such that la_2(G)=6=[(Δ+1)/2]+2,which implies that our solution is within 6 from optimal.     

Long cycles and spanning subgraphs of locally maximal 1-planar graphs

A graph is 1-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal 1-planar. For a 3-connected locally maximal 1-planar graph G, we show the existence of a spanning 3-connected planar subgraph and prove that G is Hamiltonian if G has at most three 3-vertex-cuts, and that G is traceable if G has at most four 3-vertex-cuts. Moreover, infinitely many nontraceable 5-connected 1-planar graphs are presented.     

Graph-like compacta: Characterizations and Eulerian loops

A compact graph-like space is a triple $\left(X,V,E\right)$, where $X$ is a compact, metrizable space, $V\subseteq X$ is a closed zero-dimensional subset, and $E$ is an index set such that $X⧹V\cong \text{◂+▸}E\times \left(0,1\right)$. New characterizations of compact graph-like spaces are given, connecting them to certain classes of continua, and to standard subspaces of Freudenthal compactifications of locally finite graphs. These are applied to characterize Eulerian graph-like compacta.