有向图和弱正则有向图补图的特征多项式的计算方法

本文解决了一般有向图的特征多项式的图论计算方法,并且给出了弱正则有向图和它的补图之间特征多项式的关系,从而也就解决了非负整数方阵特征多项式的图论计算方法问题。     

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.     

图论在离散事件动态系统中的应用

运用图论方法和极大代数方法,研究了非强连通图中的强连通分支的最大圈长平均值与该图的赋权邻接矩阵的特征值之间的关系,并进一步证明了其等价性.     

校园最佳游览路线问题的数学模型分析

将某高校的校园示意图转化为赋权连通图，求得该连通图的邻接矩阵，利用Floyd算法及图论软件包构造一个最短路径矩阵，得到一个赋权完全图，将求校园最佳游览路线问题归结为图论中的最佳推销员回路问题，建立混合整数线性规划模型，并利用优化软件求得最优解．从而解决了校园开放日游览计划中提出的关于校园最佳游览路线和校园游览车最优配置问题．     

战争地区邮递员问题

本文首次提出了赋权有向图上中国邮递员问题的一个推广－战争地区邮递员问题，并对解的存在性给出了若干充分条件和必要条件，得到了求解该问题的一个多项式算法。     

美国芝加哥大学中学数学设计（UCSMP）教材介绍高中第六册图论与网络一章（续）

美国芝加哥大学中学数学设计（ＵＣＳＭＰ）教材介绍高中第六册图论与网络一章（续）王申怀（北京师范大学１００８７５）§５矩阵的幂与路径矩阵Ａ中第ｉ行第ｊ列的元素记为ａｉｊ．考虑图１３中的有向图Ｇ以及它的邻接矩阵．我们规定路径中边的数目为这路径的长度，简称...     

图的特征多项式的若干性质

图的特征多项式有许多性质，本文给出了特征多项式的指数表达式，特征多项式的导数的几个不同表达式以及高阶导数的图论意义。     

有向图反能量的一个注记

有向图的反能量是指有向图的反邻接矩阵的能量.本文利用有向图的运算构造出了几类有向图,它们中的每一个都满足有向图的反能量等于其底图的能量.部分回答了Adiga等人在文[The skew energy of a digraph,Linear Algebra Appl.,2010,432:1825-1835]中提出的一个公开问题.     

具有g个零强分支的有向图的传递指数

研究了传递矩阵的图论,及布尔矩阵幂的若干图论性质,给出了有向图(布尔矩阵)传递指数的上、下界估计,从而改进了已有的结果.     

(0,1)-矩阵的积和式的图表示及其相关性质

将(0，1)．矩阵的积和式的记数问题转化为它的伴随图或伴随有向图上相关元素的记数问题，能使复杂的计数问题变得相对直观化和简单化．本文给出了(0，1)-矩阵的积和式的图论表达式，并以该表达式为基础，主要解决了2．正则图类的邻接矩阵的最大积和式的记数问题以及它的反问题，即确定了零积和式临界图的极大边数及其图类．     

A matrix-based approach to searching colored paths in a weighted colored multidigraph

An algebraic approach to finding all edge-weighted-colored paths within a weighted colored multidigraph is developed. Generally, the adjacency matrix represents a simple digraph and determines all paths between any two vertices, and is not readily extendable to colored multidigraphs. To bridge the gap, a conversion function is proposed to transform the original problem of searching edge-colored paths in a colored multidigraph to a standard problem of finding paths in a simple digraph. Moreover, edge weights can be used to represent some preference attribute. Its potentially wide realm of applicability is illustrated by a case study: status quo analysis in the graph model for conflict resolution. The explicit matrix function is more convenient than other graphical representations for computer implementation and for adapting to other applications. Additionally, the algebraic approach reveals the relationship between a colored multidigraph and a simple digraph, thereby providing new insights into algebraic graph theory.     

Bartholdi zeta and L-functions of weighted digraphs, their coverings and products

Since a zeta function of a regular graph was introduced by Ihara [Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, J. Math. Soc. Japan 19 (1966) 219-235], many kinds of zeta functions and L-functions of a graph or a digraph have been defined and investigated. Most of the works concerning zeta and L-functions of a graph contain the following: (1) defining a zeta function, (2) defining an L-function associated with a (regular) graph covering, (3) providing their determinant expressions, and (4) computing the zeta function of a graph covering and obtaining its decomposition formula as a product of L-functions. As a continuation of those works, we introduce a zeta function of a weighted digraph and an L-function associated with a weighted digraph bundle. A graph bundle is a notion containing a cartesian product of graphs and a (regular or irregular) graph covering. Also we provide determinant expressions of the zeta function and the L-function. Moreover, we compute the zeta function of a weighted digraph bundle and obtain its decomposition formula as a product of the L-functions.     

Arc transitive covering digraphs and their eigenvalues

We prove that every finite regular digraph has an arc-transitive covering digraph (whose arcs are equivalent under automorphisms) and every finite regular graph has a 2-arc-transitive covering graph. As a corollary, we sharpen C. D. Godsil's results on eigenvalues and minimum polynomials of vertex-transitive graphs and digraphs. Using Godsil's results, we prove, that given an integral matrix A there exists an arc-transitive digraph X such that the minimum polynomial of A divides that of X. It follows that there exist arc-transitive digraphs with nondiagonalizable adjacency matrices, answering a problem by P. J. Cameron. For symmetric matrices A, we construct a 2-arc-transitive graphs X.     

工件间具有串并有向图约束的单机随机排序问题

研究机器发生随机故障的单机排序问题,其中工件间的优先约束为串并有向图,目标函数为极小化加权完工时间和,证明了此问题多项式时间可解,并给出了多项式时间算法.     

New proofs for Levine’s theorems

Recently, Levine [9] expressed the vertex weighted complexity on spanning trees (with a fixed root) of the directed line graph of a digraph D in terms of the edge weighted complexity on spanning trees (with a fixed root) of D. We present new proofs for two Levine’s Theorems. Furthermore, we express the number of unicycles of the directed line graph of a digraph D in terms of the number of unicycles of D.     

On extremal weighted digraphs with no heavy paths

Bollobás and Scott proved that if the weighted outdegree of every vertex of an edge-weighted digraph is at least 1, then the digraph contains a (directed) path of weight at least 1. In this note we characterize the extremal weighted digraphs with no heavy paths. Our result extends a corresponding theorem of Bondy and Fan on weighted graphs. We also give examples to show that a result of Bondy and Fan on the existence of heavy paths connecting two given vertices in a 2-connected weighted graph does not extend to 2-connected weighted digraphs.     

Finding an induced subdivision of a digraph

We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digraph) H, does it contain an induced subdivision of a prescribed digraph D? The complexity of this problem depends on D and on whether H is an oriented graph or contains 2-cycles. We announce a number of examples of polynomial instances as well as several NP-completeness results.