几类图的匹配多项式之间的关系与一类图的匹配等价图

研究了几类图的匹配多项式以及它们之间的一些整除关系,给出了路的匹配多项式相互整除的一个充分必要条件,并且刻画了图T2,2,n的所有匹配等价图.     

几乎等长T形树的匹配唯一性

讨论简单无向图G的匹配唯一性,研究T形树T(m,n,s)匹配唯一的充分条件.利用匹配多项式根的信息,根据其定义以及图的度序列和匹配多项式的性质推导.若T形树T(m,n,s)是几乎等长的,则其是匹配唯一的.找到了T形树T(m,n,s)匹配唯一的一个充分条件,并得到了图的匹配多项式根的一些性质.     

T(2,2,n)∪(∪i∈A C_i)的匹配等价图类

利用图的匹配多项式及其最大实数根的性质完整刻画了T(2,2,n)∪(∪i∈A Ci)(n≥3,A是大于等于3的整数组成的可重集)的匹配等价图类.     

T(2,2,n) U (U Ci)的匹配等价图类i∈A

利用图的匹配多项式及其最大实数根的性质完整刻画了T(2,2,n)U(U C<,t>)(n≥3,A是大于等于3的整数组成的可重集)的匹配等价图类.     

饱和链烃的Hosoya指数与其分子结构的匹配多项式及根

一个图的Hosoya指数实际上是该图的匹配多项式的系数之和.我们计算了40种饱和链烃的Hosoya指数和其分子结构图的匹配多项式以及匹配多项式的最大根,最后经过通过SPSS10.0软件进行曲线拟合发现各饱和链烃的沸点与Hosoya指数具有对数函数的关系,如下：其沸点b.p.（℃）=-98.674＋64.5488＊ln（...     

K_(1,1,m)□P_n的交叉数

借助拉链积运算,Cartesian积图K(1,m)□Pn和K(2,m)□Pn的交叉数最近被先后确定.本文进一步证明了:对于m,n≧1,有cr(K(1,1,m)□Pn)=2n[m/2][(m-1)/2]+(n-1)[m/2].结论的证明基于Bokal关于树的Cartesian积图交叉数的有关结果.另外,我们也给出了确定K(2,m)□Pn交叉数的一个简洁方法.     

图的不可靠性多项式的递推公式

本文得到了图的不可靠性多项式的一个递推关系,然后利用这个关系求出完全图,星、完全图与空图的连图、完全二部图、路以及圈等一些特殊图类的不可靠性多项式.     

图的匹配与拉普拉斯特征值

设G=(V(G),E(G))是一个图,M是E(G)的—个子集.如果M中任意两条边均无公共端点,则称M为图G的匹配.如果图G的一个匹配M中的边恰好关联G的每一个顶点,则称M为图G的完美匹配.如果图G中除了一个顶点以外,其他所有顶点都与匹配M中的边相关联,则称M为图G的几乎完美匹配.如果对任意v∈V(G), G-v均有完美匹配,则称G是因子临界的.本文中,我们给出了判定一个图有完美匹配、或者几乎完美匹配或者是因子临界的拉普拉斯谱条件.     

图的两类一般多项式

<正> 本文提出图的边覆盖多项式;它与 Farrell 提出的 F-多项式(点覆盖多项式)一起,可以概括图的若干重要多项式,如特征多项式、色多项式、树多项式、匹配多项式、结构多项式及本文的 Euler 多项式等.我们就这两类多项式,证明了两个一般形式的消去定理.然后,将赋权图及有向图的几种多项式纳入此框架之中,从而得到有关行列式、承袭式、特征多项式及赋权匹配多项式的若干结果.     

关于分数k-因子临界图与分数k-可扩图的若干结果

一个简单图G, 如果对于V(G)的任意k元子集S, 子图G-S都包含分数完美匹配, 那么称G为分数k-因子临界图. 如果图G的每个k-匹配M都包含在一个分数完美匹配中, 那么称图G为分数k-可扩图. 给出一个图是分数k-因子临界图和分数k-可扩图的充分条件, 并给出一个图是分数k-因子临界图的充分必要条件.     

Maximum matching and the game of Slither

Slither is a game played on a finite graph in which the players alternately choose edges so as to form a path. In this paper we present a strategy for Slither. The strategy depends upon an application of Edmond's maximum matching algorithm to the graph and a sequence of induced subgraphs. The strategy is practical in the sense that the amount of computation necessary has a polynomial bound.     

On the matching polynomial of subdivision graphs

Let G be a simple graph and let S(G) be the subdivision graph of G, which is obtained from G by replacing each edge of G by a path of length two. In this paper, by the Principle of Inclusion and Exclusion we express the matching polynomial and Hosoya index of S(G) in terms of the matchings of G. Particularly, if G is a regular graph or a semi-regular bipartite graph, then the closed formulae of the matching polynomial and Hosoya index of S(G) are obtained. As an application, we prove a combinatorial identity.     

On the Location of Roots of Graph Polynomials

Roots of graph polynomials such as the characteristic polynomial, the chromatic polynomial, the matching polynomial, and many others are widely studied. In this paper we examine to what extent the location of these roots reflects the graph theoretic properties of the underlying graph.     

两类树的独立集多项式的单峰性

研究了图的独立集多项式的单峰性,给出具有爪图结构的几类图的独立集多项式等价的无爪图,并在此基础上证明了两类具有爪图结构的树T(n,n+1,m)和T(I,i+1,k,j,j+1)的独立集多项式具有单峰性,从而为具有爪图结构的其它树的单峰性提供了一个证明方法.     

Multi-criteria approximation schemes for the resource constrained shortest path problem

In the resource constrained shortest path problem we are given a directed graph along with a source node and a destination node, and each arc has a cost and a vector of weights specifying its requirements from a set of resources with finite budget limits. A minimum cost source-destination path is sought such that the total consumption of the arcs from each resource does not exceed its budget limit. In the case of constant number of weight functions we give a fully polynomial time multi-criteria approximation scheme for the problem which returns a source-destination path of cost at most the optimum, however, the path may slightly violate the budget limits. On the negative side, we show that there does not exist a polynomial time multi-criteria approximation scheme for the problem if the number of weight functions is not a constant. The latter result applies to a broad class of problems as well, including the multi-dimensional knapsack, the multi-budgeted spanning tree, the multi-budgeted matroid basis and the multi-budgeted bipartite perfect matching problems.     

Higher-order matching polynomials and d-orthogonality

We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials—the Chebyshev, Hermite, and Laguerre polynomials—can be interpreted as matching polynomials of paths, cycles, complete graphs, and complete bipartite graphs. The notion of d-orthogonality is a generalization of the usual idea of orthogonality for polynomials and we use sign-reversing involutions to show that the higher-order Chebyshev (first and second kinds), Hermite, and Laguerre polynomials are d-orthogonal. We also investigate the moments and find generating functions of those polynomials.     

On Path Factors of (3, 4)-Biregular Bigraphs

A (3, 4)-biregular bigraph G is a bipartite graph where all vertices in one part have degree 3 and all vertices in the other part have degree 4. A path factor of G is a spanning subgraph whose components are nontrivial paths. We prove that a simple (3,4)-biregular bigraph always has a path factor such that the endpoints of each path have degree three. Moreover we suggest a polynomial algorithm for the construction of such a path factor.     

Paths, trees and matchings under disjunctive constraints

We study the minimum spanning tree problem, the maximum matching problem and the shortest path problem subject to binary disjunctive constraints: A negative disjunctive constraint states that a certain pair of edges cannot be contained simultaneously in a feasible solution. It is convenient to represent these negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the edges of the underlying graph, and whose edges encode the constraints.We prove that the minimum spanning tree problem is strongly NP-hard, even if every connected component of the conflict graph is a path of length two. On the positive side, this problem is polynomially solvable if every connected component is a single edge (that is, a path of length one). The maximum matching problem is NP-hard for conflict graphs where every connected component is a single edge.Furthermore we will also investigate these graph problems under positive disjunctive constraints: In this setting for certain pairs of edges, a feasible solution must contain at least one edge from every pair. We establish a number of complexity results for these variants including APX-hardness for the shortest path problem.     

A note on the matching numbers of triangle-free graphs

For the sake of brevity the absolute values of the coefficients of the matching polynomial of a graph are called matching numbers in this note. It is shown that for a triangle-free graph these numbers coincide with the coefficients of the chromatic polynomial of its complement when this polynomial is written in factorial form. As an application it is mentioned that the coefficients of every rook polynomial are at the same time coefficients of some chromatic polynomial.     

Unicycle graphs and uniquely restricted maximum matchings

A matching M is called uniquely restricted in a graph G if it is the unique perfect matching of the subgraph induced by the vertices that M saturates. G is a unicycle graph if it owns only one cycle. Golumbic, Hirst and Lewenstein observed that for a tree or a graph with only odd cycles the size of a maximum uniquely restricted matching is equal to the matching number of the graph. In this paper we characterize unicycle graphs enjoying this equality. Moreover, we describe unicycle graphs with only uniquely restricted maximum matchings. Using these findings, we show that unicycle graphs having only uniquely restricted maximum matchings can be recognized in polynomial time.