n阶三圈图的补图的谱半径

三圈图是边数等于顶点数加2的简单连通图.在所有n阶三圈图的补图中,哪一个的谱半径最大?文中给出了n阶三圈图的补图的谱半径的上界,并刻画了唯一的达到该上界的图.     

围长为g且有k个悬挂点的单圈图的谱半径和Laplacian谱半径

图的谱半径和Laplacian谱半径分别是图的邻接矩阵和Laplacian矩阵的最大特征值.本文中,我们分别刻画了围长为g且有k个悬挂点的单圈图的谱半径和Laplacian谱半径达到最大时的极图.     

有向图的Laplace谱半径

Laplace矩阵的谱半径一直是近年来谱图理论的研究热点.本文主要讨论有向图Laplace矩阵的谱半径,用顶点的出度和公共邻域数给出了谱半径上界,用图的最大出度给出了一些特殊图类谱半径的下界.     

一致超图谱半径界的改进结果

利用张量理论研究一致超图的谱半径.首先,利用对角相似张量与原张量同谱的性质,结合张量特征值的圆盘定理,给出谱半径的上界,这一上界严格小于最大度;其次,通过超图的度向量给出谱半径的下界.改进了超图谱半径上下界的原有结果.     

连通图的最小Laplace谱半径的排序

首先找出了具有最小Laplace谱半径的第2个至第5个n阶单圈图和具有最小Laplace谱半径的n阶双圈图.然后结合有关n阶树的最小Laplace谱半径的排序,给出了所有n阶连通图中Laplace谱半径最小的14个图,当n为偶数时,它们达到了所有佗阶连通图中Laplace谱半径最小的9个值(其中有并列的),而当n为奇数时,它们则达到了Laplace谱半径最小的8个值(其中有并列的).     

具有固定围长的单圈图的无号拉普拉斯谱半径

研究了单圈图的无号拉普拉斯谱半径,给出了具有固定围长的单圈图的无号拉普拉斯谱半径最大的图.     

矩阵谱半径上界的估值及非负矩阵谱半径的一个公式

<正> 本文建立了循环矩阵和非负矩阵谱半径的公式,并提出几个不等式.用这些不等式估计矩阵谱半径的上界,可得到比一般方法更精确的估计,把这些不等式作为矩阵敛散的判据,则可得到比[2]、[3]更精确、应用范围更广的结果.由于估出了谱半径的上界,故能了解矩阵特征值分布的区域.对于估计循环矩阵谱半径的上界,我们提出了一个比较精确的公式,它有时能定出循环矩阵谱半径的上确界.     

关于k树的谱半径

洪渊给出了谱半径最大的k树.该文进一步定义了关于k树的一个参数l(G),借之给出了谱半径达到第二大和第三大的k树.     

关于图的拟拉普拉斯谱半径

用代数方法给出了一个关于简单图的顶点度数与拟拉普拉斯谱半径的不等式,并给出了图的拟拉普拉斯谱半径的一个新上界.     

强连通双圈有向图的无符号拉普拉斯谱刻画

设$\overrightarrow{G}$ 是一个强连通双圈有向图, $A(\overrightarrow{G})$是其邻接矩阵.设$D(\overrightarrow{G})$ 是$\overrightarrow{G}$的顶点出度的对角矩阵, $Q(\overrightarrow{G})=D(\overrightarrow{G})+A(\overrightarrow{G})$是$\overrightarrow{G}$ 的无符号拉普拉斯矩阵. $Q(\overrightarrow{G})$的谱半径称为$\overrightarrow{G}$的无符号拉普拉斯谱半径.在这篇文章中, 确定了在所有强连通双圈有向图中达到最大或最小无符号拉普拉斯谱半径的唯一有向图. 此外,还证明了任意一个强连通双圈有向图是由它的无符号拉普拉斯谱所确定的.     

Realization of primitive branched coverings over closed surfaces following the hurwitz approach

Let V be a closed surface, H⊑π₁(V) a subgroup of finite index l and D=[A ₁,...,A _m ] a collection of partitions of a given number d≥2 with positive defect v(D). When does there exist a connected branched covering f:W→V of order d with branch data D and f∶W→V It has been shown by geometric arguments [4] that, for l=1 and a surface V different from the sphere and the projective plane, the corresponding branched covering exists (the data D is realizable) if and only if the data D fulfills the Hurwitz congruence v(D)э0 mod 2. In the case l>1, the corresponding branched covering exists if and only if v(D)э0 mod 2, the number d/l is an integer, and each partition A _i ∈D splits into the union of l partitions of the number d/l. Here we give a purely algebraic proof of this result following the approach of Hurwitz [11]. The realization problem for the projective plane and l=1 has been solved in [7,8]. The case of the sphere is treated in [1, 2, 12, 7].     

On compactness of the difference of composition operators

Let φ and ψ be analytic self-maps of the unit disc, and denote by C_φ and C_ψ the induced composition operators. The compactness and weak compactness of the difference T=C_φ−C_ψ are studied on H^p spaces of the unit disc and L^p spaces of the unit circle. It is shown that the compactness of T on H^p is independent of p∈[1,∞). The compactness of T on L¹ and M (the space of complex measures) is characterized, and examples of φ and ψ are constructed such that T is compact on H¹ but non-compact on L¹. Other given results deal with L^∞, weakly compact counterparts of the previous results, and a conjecture of J.E. Shapiro.     

Disjunctive Rado numbers

If L₁ and L₂ are linear equations, then the disjunctive Rado number of the set {L₁,L₂} is the least integer n, provided that it exists, such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to either L₁ or L₂. If such an integer n does not exist, then the disjunctive Rado number is infinite. In this paper, it is shown that for all integers and b1, the disjunctive Rado number for the equations x₁+a=x₂ and x₁+b=x₂ is a+b+1-gcd(a,b) if is odd and the disjunctive Rado number for these equations is infinite otherwise. It is also shown that for all integers a>1 and b>1, the disjunctive Rado number for the equations ax₁=x₂ and bx₁=x₂ is c^s+t-1 if there exist natural numbers c,s, and t such that a=c^s and b=c^t and s+t is an odd integer and c is the largest such integer, and the disjunctive Rado number for these equations is infinite otherwise.     

Some paranormed sequence spaces of non-absolute type derived by weighted mean

The sequence spaces ?_∞(p), c(p) and c₀(p) were introduced and studied by Maddox [I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64 (1968) 335-340]. In the present paper, the sequence spaces λ(u,v;p) of non-absolute type which are derived by the generalized weighted mean are defined and proved that the spaces λ(u,v;p) and λ(p) are linearly isomorphic, where λ denotes the one of the sequence spaces ?_∞, c or c₀. Besides this, the β- and γ-duals of the spaces λ(u,v;p) are computed and the basis of the spaces c₀(u,v;p) and c(u,v;p) is constructed. Additionally, it is established that the sequence space c₀(u,v) has AD property and given the f-dual of the space c₀(u,v;p). Finally, the matrix mappings from the sequence spaces λ(u,v;p) to the sequence space μ and from the sequence space μ to the sequence spaces λ(u,v;p) are characterized.     

Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance

The polynomial birth–death distribution (abbreviated, PBD) on ℐ={0,1,2,…} or ℐ={0,1,2,…,m} for some finite m introduced in Brown and Xia (Ann. Probab. 29:1373–1403, 2001) is the equilibrium distribution of the birth–death process with birth rates {α _i } and death rates {β _i }, where α _i ≥0 and β _i ≥0 are polynomial functions of i∈ℐ. The family includes Poisson, negative binomial, binomial, and hypergeometric distributions. In this paper, we give probabilistic proofs of various Stein’s factors for the PBD approximation with α _i =a and β _i =i+bi(i−1) in terms of the Wasserstein distance. The paper complements the work of Brown and Xia (Ann. Probab. 29:1373–1403, 2001) and generalizes the work of Barbour and Xia (Bernoulli 12:943–954, 2006) where Poisson approximation (b=0) in the Wasserstein distance is investigated. As an application, we establish an upper bound for the Wasserstein distance between the PBD and Poisson binomial distribution and show that the PBD approximation to the Poisson binomial distribution is much more precise than the approximation by the Poisson or shifted Poisson distributions.      

Positive Derivations on Archimedean Almost f-Rings

It is shown by P. Colville, G. Davis and K. Keimel that if R is an Archimedean f-ring then a positive group endomorphism D on R is a derivation if and only if the range of D is contained in N(R) and the kernel of D contains R ², where N(R) is the set of all nilpotent elements in R and R ² is the set of all products uv in R. The main objective of this paper is to establish the result corresponding to the Colville–Davis–Keimel theorem in the almost f-ring case. The result obtained in this regard is that if D is a positive derivation in an Archimedean almost f-ring, then the range of D is contained in N(R) and the kernel of D contains R ³, where R ³ is the set of all products uvw in R. Examples are produced showing that, contrary to the f-ring case, the converse is in general false and the third power is the best possible.     

Analytical relationships between metric and centrality measures of a network and its dual

The centrality and efficiency measures of a network G are strongly related to the respective measures on the dual G^? and the bipartite B(G) associated networks. We show some relationships between the Bonacich centralities c(G), c(G^?) and c(B(G)) and between the efficiencies E(G) and E(G^?) and we compute the behavior of these parameters in some examples.     

On reformulated Zagreb indices

The first and second reformulated Zagreb indices are defined respectively in terms of edge-degrees as EM₁(G)=∑_e∈Edeg(e)² and EM₂(G)=∑_e∼fdeg(e)deg(f), where deg(e) denotes the degree of the edge e, and e∼f means that the edges e and f are adjacent. We give upper and lower bounds for the first reformulated Zagreb index, and lower bounds for the second reformulated Zagreb index. Then we determine the extremal n-vertex unicyclic graphs with minimum and maximum first and second Zagreb indices, respectively. Furthermore, we introduce another generalization of Zagreb indices.     

Waiting Time Problems in a Two-State Markov Chain

Let F ₀ be the event that l ₀ 0-runs of length k ₀ occur and F ₁ be the event that l ₁ 1-runs of length k ₁ occur in a two-state Markov chain. In this paper using a combinatorial method and the Markov chain imbedding method, we obtained explicit formulas of the probability generating functions of the sooner and later waiting time between F ₀ and F ₁ by the non-overlapping, overlapping and "greater than or equal" enumeration scheme. These formulas are convenient for evaluating the distributions of the sooner and later waiting time problems.     

A phase transition in the random transposition random walk

Our work is motivated by Bourque and Pevzner's (2002) simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk on the group of permutations on n elements. Consider this walk in continuous time starting at the identity and let D _t be the minimum number of transpositions needed to go back to the identity from the location at time t. D _t undergoes a phase transition: the distance D _{cn /2}∼u(c)n, where u is an explicit function satisfying u(c)=c/2 for c≤1 and u(c)<c/2 for c>1. In addition, we describe the fluctuations of D _{cn /2} about its mean in each of the three regimes (subcritical, critical and supercritical). The techniques used involve viewing the cycles in the random permutation as a coagulation-fragmentation process and relating the behavior to the Erdős-Renyi random graph model.