On Finite Groups with a Given Number of Centralizers

For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G) = |Cent(G)|. G is called an n-centralizer group if #Cent(G) = n, and a primitive n-centralizer group if #Cent(G) = #Cent(G/Z(G)) = n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n 2, 3, there exists a finite group G with #Cent(G) = n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G) = 6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) A₄, the alternating group on four letters. Also, we prove that, if G/Z(G) A₄, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) A₄ and #Cent(G) = 8.This research was in part supported by a grant from IPM.2000 Mathematics Subject Classification: 20D99, 20E07     

Trees with Given Stability Number and Minimum Number of Stable Sets

We study the structure of trees minimizing their number of stable sets for given order n and stability number α. Our main result is that the edges of a non-trivial extremal tree can be partitioned into n ? α stars, each of size \({\lceil\frac{n-1}{n-\alpha}\rceil}\) or \({\lfloor\frac{n-1}{n-\alpha}\rfloor}\) , so that every vertex is included in at most two distinct stars, and the centers of these stars form a stable set of the tree.     

关于具有给定非悬挂边数的树的极小能量

图的能量定义为其特征的绝对值之和.Γ(n,q)表示所有具有n个顶点,q条非悬挂边的树构成的集合.本文中,我们利用两个变换确定了Γ(n,q)中具有极小、第二小能量的树.     

Factor-Critical Graphs with Given Number of Maximum Matchings

A connected graph G is said to be factor-critical if G − ν has a perfect matching for every vertex ν of G. In this paper, the factor-critical graphs G with |V(G)| maximum matchings and with |V(G)| + 1 ones are characterized, respectively. From this, some special bicritical graphs are characterized. This work is supported by the Ph.D. Programs Foundation of Ministry of Education of China (No.20070574006) and the NNSF(10201019) of China.     

Generalized Multiresolution Analyses with Given Multiplicity Functions

Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space ℋ that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed scaling function. Previous authors have studied a multiplicity function m which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space ℋ is L ²(ℝ ⁿ ), the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function m satisfying a consistency condition, which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function m.     

Construction of Separately Continuous Functions with Given Restriction

We solve the problem of the construction of separately continuous functions on a product of two topological spaces with given restriction. It is shown, in particular, that, for an arbitrary topological space X and a function g: X R of the first Baire class, there exists a separately continuous function f: X × X R such that f(x, x) = g(x) for every x X.     

Extremal Zagreb Indices of Graphs with a Given Number of Cut Edges

For a graph, the first Zagreb index M ₁ is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M ₂ is equal to the sum of the products of the degrees of pairs of adjacent vertices. Denote by ${\mathcal{G}_{n,k}}$ the set of graphs with n vertices and k cut edges. In this paper, we showed the types of graphs with the largest and the second largest M ₁ and M ₂ among ${\mathcal{G}_{n,k}}$ .     

Growth of Entire Functions with Given Zeros and Representation of Meromorphic Functions

Mathematical Notes - Let $\Lambda = \{ \lambda n\}$ be a sequence of points on the complex plane, and let $\Lambda (r)$ be the number of points of the sequence $\Lambda$ in the disk $\{...     

Degree Distance of Unicyclic Graphs with Given Matching Number

Let G be a connected graph with vertex set V(G). The degree distance of G is defined as ${D'(G) = \sum_{\{u, v\}\subseteq V(G)} (d_G(u) + d_G (v))\, d(u,v)}$ , where d _G (u) is the degree of vertex u, d(u, v) denotes the distance between u and v, and the summation goes over all pairs of vertices in G. In this paper, we characterize n-vertex unicyclic graphs with given matching number and minimal degree distance.     

Extremal Graphs With a Given Number of Perfect Matchings

Let denote the maximum number of edges in a graph having n vertices and exactly p perfect matchings. For fixed p, Dudek and Schmitt showed that for some constant when n is at least some constant . For , they also determined and . For fixed p, we show that the extremal graphs for all n are determined by those with vertices. As a corollary, a computer search determines and for . We also present lower bounds on proving that for (as conjectured by Dudek and Schmitt), and we conjecture an upper bound on . Our structural results are based on Lovász's Cathedral Theorem.     

On the Characterization of Maximal Planar Graphs with a Given Signed Cycle Domination Number

Let G =(V, E) be a simple graph. A function f : E → {+1,-1} is called a signed cycle domination function(SCDF) of G if ∑_(e∈E(C))f(e) ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γ'_(sc)(G) = min{∑_(e∈E)f(e)| f is an SCDF of G}. This paper will characterize all maximal planar graphs G with order n ≥ 6 and γ'_(sc)(G) = n.     

The Algebraic Connectivity of Graphs with Given Matching Number

The algebraic connectivity of a graph is the second smallest eigenvalue of the associated Laplacian matrix. In this paper, we not only characterize the extremal graphs with the maximal algebraic connectivity among all graphs of order n with given matching number, but also determine the extremal tree with the maximal algebraic connectivity among all trees of order n with given matching number.     

The Size of a Maximum Subgraph of the Random Graph with a Given Number of Edges

Doklady Mathematics - We have proven that the maximum size k of an induced subgraph of the binomial random graph $$G(n,p)$$ with a given number of edges $$e(k)$$ (under certain conditions on this...     

非幂零极大子群共轭类类数给定的有限群

证明了非幂零极大子群共轭类类数等于2的有限群必可解,并给出了非幂零极大子群同阶类类数等于2的非可解群的等价刻画.     

给定零点的指数型整函数Ⅱ

研究了如下问题:给定右半平面一复数列,以该复数列为零点的指数型整函数在虚轴上的增长性有什么表现?并给出了完整的解答.这是对Malliavin和Rubel关于给定右半平面一复数列为零点的指数型整函数性质的研究工作的一个推广.     

The Number of Subtrees of Trees with Given Degree Sequence

This article investigates some properties of the number of subtrees of a tree with given degree sequence. These results are used to characterize trees with the given degree sequence that have the largest number of subtrees, which generalize the recent results of Kirk and Wang (SIAM J Discrete Math 22 (2008), 985–995). These trees coincide with those which were proven by Wang and independently Zhang et al. (2008) to minimize the Wiener index. We also provide a partial ordering of the extremal trees with different degree sequences, some extremal results follow as corollaries.