具有完美匹配树的代数连通度的排序

Jason等确定了阶数为n的具有完美匹配树的最大的代数连通度以及相应的极图．本文确定了阶数为n的具有完美匹配树的第二大到第五大的代数连通度以及达到这些数值的图（或图类）．     

关于树的代数连通度的Fiedler不等式的新证明

设T为含n个顶点的树，L(T)为其Laplace矩阵，L(T)的次小特征值α(T)称为T的代数连通度，Fiedlcr给出如下关于α(T)的界的经典结论α(Pn)≤α(T)≤α(Sn)，其中Pn，Sn分别为含有n个顶点的路和星．Merris和Mass独立地证明了：α(T)=α(Sn)当且仅当T=Sn．通过重新组合由Fiedler向量所赋予的顶点的值，本给出上述不等式的新证明，并证明了：α(T)=α(Pn)当且仅当T=Pn。     

关于代数连通度的一个注记

图G=(V,E)的次小的拉普拉斯特征值称为G的代数连通度,记为α(G).设δ(G)为G的最小度.Fiedler早在1973年便证明了α(G)≤δ(G),但他未能给出等号成立的极图刻划.后来,我们在[6]中确定了当δ(G)≤1/2|V(G)|时α(G)=δ(G)的充要条件.本文中,我们将确定任意情况下α(G)=δ(G)成立的所有极图.     

树的变形与代数连通度

本文利用瓶颈矩阵的Perron值和代数连通度的二次型形式,系统地研究了当迁移或改变分支(边、点)和变动一些边的权重时无向赋权树的代数连通度的变化规律,认为代数连通度可用来描述树的边及其权重的某种中心趋势性.引入广义树和广义特征点概念,将II型树转换成具有相同代数连通度的I型树,使得树的代数连通度的讨论只须限于I型树的研究即可.     

Perfect Matchings of Cellular Graphs

We introduce a family of graphs, called cellular, and consider the problem of enumerating their perfect matchings. We prove that the number of perfect matchings of a cellular graph equals a power of 2 times the number of perfect matchings of a certain subgraph, called the core of the graph. This yields, as a special case, a new proof of the fact that the Aztec diamond graph of order n introduced by Elkies, Kuperberg, Larsen and Propp has exactly 2 ^n(n+1)/2 perfect matchings. As further applications, we prove a recurrence for the number of perfect matchings of certain cellular graphs indexed by partitions, and we enumerate the perfect matchings of two other families of graphs called Aztec rectangles and Aztec triangles.     

The effect on the algebraic connectivity of a tree by grafting or collapsing of edges

Let G=(V,E) be a tree on n?2 vertices and let v∈V. Let L(G) be the Laplacian matrix of G and μ(G) be its algebraic connectivity. Let G_k,l, be the graph obtained from G by attaching two new paths P:vv₁v₂…v_k and Q:vu₁u₂…u_l of length k and l, respectively, at v. We prove that if l?k?1 then μ(G_k-1,l+1)?μ(G_k,l). Let (v₁,v₂) be an edge of G. Let be the tree obtained from G by deleting the edge (v₁,v₂) and identifying the vertices v₁ and v₂. Then we prove that As a corollary to the above results, we obtain the celebrated theorem on algebraic connectivity which states that among all trees on n vertices, the path has the smallest and the star has the largest algebraic connectivity.     

Perfect Matchings of Regular Bipartite Graphs

Let G be a regular bipartite graph and . We show that there exist perfect matchings of G containing both, an odd and an even number of edges from X if and only if the signed graph , that is a graph G with exactly the edges from X being negative, is not equivalent to . In fact, we prove that for a given signed regular bipartite graph with minimum signature, it is possible to find perfect matchings that contain exactly no negative edges or an arbitrary one preselected negative edge. Moreover, if the underlying graph is cubic, there exists a perfect matching with exactly two preselected negative edges. As an application of our results we show that each signed regular bipartite graph that contains an unbalanced circuit has a 2‐cycle‐cover such that each cycle contains an odd number of negative edges.     

On Vertex Connectivity and Absolute Algebraic Connectivity for Graphs

Let G be a graph on n vertices with vertex connectivity v with 1 ≤ v ≤ n -2. We produce an attainable upper bound on the absolute algebraic connectivity of G in terms of n and v .     

On Vertex Connectivity and Absolute Algebraic Connectivity for Graphs

Let G be a graph on n vertices with vertex connectivity v with 1 h v h n m 2. We produce an attainable upper bound on the absolute algebraic connectivity of G in terms of n and v .     

用拉普拉斯谱半径定序给定阶和匹配数的树

Let T（n,i） be the set of all trees with order n and matching number i.We determine the third to sixth trees in T（2i ＋ 1,i） and the third to fifth trees in T（n,i） for n ≥ 2i ＋ 2 with the largest Laplacian spectral radius.     

渺位六角系统完美匹配Z——变换图

六角系统是2-连通的平面图，其每个内部面都是单位正六边形．六角系统的完美匹配是化学中苯类芳烃体系的Kekule结构．一个六角系统H完美匹配Z—变换图Z（H）是一个图，它的顶点集是H的完匹配集，两个匹配相邻当且仅当它们的对称差是一个单位正六边形．本文用乘积图刻划了沙位六角系统Z—变换图的结构．     

图的代数连通度的界(英文)

本文首先给出了简单图的度序列的平方和的上界,利用这些结果,求出了简单图的代数连通度的几个上下界并确定了它们的临界图。另外,文章也给出了加权图的代数连通度的一个下界。     

Algebraic connectivity of weighted trees under perturbation

We investigate how the algebraic connectivity of a weighted tree behaves when the tree is perturbed by removing one of its branches and replacing it with another. This leads to a number of results, for example the facts that replacing a branch in an unweighted tree by a star on the same number of vertices will not decrease the algebraic connectivity, while replacing a certain branch by a path on the same number of vertices will not increase the algebraic connectivity. We also discuss how the arrangement of the weights on the edges of a tree affects the algebraic connectivity, and we produce a lower bound on the algebraic connectivity of any unweighted graph in terms of the diameter and the number of vertices. Throughout, our techniques exploit a connection between the algebraic connectivity of a weighted tree and certain positive matrices associated with the tree.     

一类二部图的几乎完美匹配数

图 的一个 匹配称 为几 乎完美 匹配,若 它覆 盖了一 个顶点 以外 的所有 顶点． 本文 给出具 正 Ｓｕｒｐ ｌｕ ｓ二部图有 Ｖ（ Ｇ）＋ １个几乎 完美匹配的两个 充要条件     

Perfect Matchings Avoiding Several Independent Edges in a Star‐Free Graph

In Aldred and Plummer (Discrete Math 197/198 (1999) 29–40) proved that every m‐connected ‐free graph of even order has a perfect matching M with and , where F₁ and F₂ are prescribed disjoint sets of independent edges with and . It is known that if l satisfies , then the star‐free condition in the above result is best possible. In this paper, for , we prove a refinement of the result in which the condition is replaced by the weaker condition that G is ‐free (note that the new condition does not depend on l). We also show that if m is even and either or , then for m‐connected graphs G with sufficiently large order, one can replace the condition by the still weaker condition that G is ‐free. The star‐free conditions in our results are best possible.     

树按Balaban指标的排序(英文)

连通图G的Balaban指标(也称J指标)定义为J=J(G)=(|E(G)|)/μ+1∑_(uυ∈E(G)),其中σ_G(u)=∑(w∈V(G)d_G(u,w)此处μ是基圈数.Balaban指标常用于各种QSAR和QSPR的研究.本文根据Balaban指标的计算公式及文中提到的变换方式,我们得到了一些序关系.基于这些序关系,我们确定了n个顶点的树中具有最小Balaban指标的前21个树.     

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.     

Edge‐Disjoint Spanning Trees,Edge Connectivity,and Eigenvalues in Graphs

Let and denote the second largest eigenvalue and the maximum number of edge‐disjoint spanning trees of a graph G, respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of , Cioab? and Wong conjectured that for any integers and a d‐regular graph G, if , then . They proved the conjecture for , and presented evidence for the cases when . Thus the conjecture remains open for . We propose a more general conjecture that for a graph G with minimum degree , if , then . In this article, we prove that for a graph G with minimum degree δ, each of the following holds.

• (i) For , if and , then .
• (ii) For , if and , then .

Our results sharpen theorems of Cioab? and Wong and give a partial solution to Cioab? and Wong's conjecture and Seymour's problem. We also prove that for a graph G with minimum degree , if , then the edge connectivity is at least k, which generalizes a former result of Cioab?. As corollaries, we investigate the Laplacian and signless Laplacian eigenvalue conditions on and edge connectivity.     

On the Upper Bounds of the Numbers of Perfect Matchings in Graphs with Given Parameters

Let φ(G),κ(G),α(G),χ(G),cl(G),diam(G)denote the number of perfect matchings,connectivity,independence number,chromatic number,clique number and diameter of a graph G,respectively.In this note,by constructing some extremal graphs,the following extremal problems are solved:1.max{φ(G):|V(G)|=2n,κ(G)≤k}=k[(2n-3)!!],2.max{φ(G):|V(G)|=2n,α(G)≥k}=[multiply from i=0 to k-1(2n-k-i)[(2n-2k-1)!!],3.max{φ(G):|V(G)|=2n,χ(G)≤k}=φ(T_(k,2n))T_(k,2n)is the Turán graph,that is a complete k-partite graphon 2n vertices in which all parts are as equal in size as possible,4.max{φ(G):|V(G)|=2n,cl(G)=2}=n1,5.max{φ(G):|V(G)|=2n,diam(G)≥2}=(2n-2)(2n-3)[(2n-5)!!],max{φ(G):|V(G)|=2n,diam(G)≥3}=(n-1)~2[(2n-5)!!].