2-控制数和连通2-控制数相等的图(英文)

任意一个图G =(V ,E) ,S是V(G)的子集 ,如果对每一个顶点u∈V-S都存在顶点v∈S ,使得d(u ,v) ≤ 2 ,则称S为G的一个 2 控制 .称最小的 2 控制集的顶点个数为G的 2 控制数 ,记为γ2 (G) .如果G的一个 2 控制集S的生成子集〈S〉是一个连通图 ,则称S为G的一个连通 2 控制集 .称最小的连通 2 控制集的顶点个数为G的连通 2 控制数 ,记为γc2 (G) .本文论述了树和单圈图中 2 控制数和连通 2 控制数相等的充分必要条件 .     

匹配数为2的图的最小特征值

在所有给定阶数且匹配数为2的连通图中,我们刻画了最小特征值达到极小的图,给出了这类图最小特征值的下界.     

补图具有悬挂点且连通的图的最小特征值

图的最小特征值定义为图的邻接矩阵的最小特征值,它是刻画图的结构性质的重要参数.在给定阶数且补图为具有悬挂点的连通图的图类中,刻画了最小特征值达极小的唯一图,并给出了这类图最小特征值的下界.     

恰有两个主特征值的图与图的剖分

大量研究表明,图的主特征值的数量与图的结构有着密切关系.通过恰有两个主特征值的图的特征定义了2-邻域k-剖分图,研究了恰有两个主特征值的图与2-邻域k-剖分图之间的关系;同时给出一个2-邻域k-剖分图在k=2,3时为等部剖分的条件.     

最大2-正则诱导子图的长度

设G是2-连通图，c(G)是图G的最长诱导圈的长度，c′(G)是图G的最长诱导2-正则子图的长度。本文我们用图的特征值给出了c(G)和c′(G)的几个上界。     

关于连通度固定的图的拉普拉斯谱半径的一个注记(英文)

图的拉普拉斯谱半径是其拉普拉斯矩阵的最大特征值.本文刻画了(边)连通度至多为k的二部图中具有最大拉普拉斯谱半径的所有图.[Linear Algebra Appl.,2009,431(1):99-103]也考虑了此问题,而所得到的结果并不完整.     

电阻-距离谱的若干结果

对连通图$G$的顶点$u$和$v$, $u$与$v$在$G$中的电阻距离$r_G(u,v)$等于相邻顶点之间的电阻为单位电阻的$G$对应的电网中$u$与$v$之间的等效电阻. 图$G$的电阻-距离特征值是$G$的电阻-距离矩阵$R(G)=(r_G(u,v))_{u,v\in V(G)}$的特征值. 我们分别确定了不同于完全图与完全图删去一条边后得到的图及给定割边数目的使得最大电阻-距离特征值取得最小值的唯一的连通图, 还讨论了最小电阻-距离特征值的性质.     

2-型模糊环境下的能源系统优化

基于模糊可能性理论,建立2-型模糊环境下的能源分配优化模型,其中各种类型能源的成本用2-型模糊变量刻画.用均值简约方法简约2-型模糊成本,建立广义期望值意义下的模糊能源分配优化模型.当成本用相互独立的三角2-型模糊变量刻画时,所建立的模糊能源分配优化模型可以转化为等价的参数线性规划.最后提供一个数值例子表明建模思想.     

极小正则2-图及其应用

2-图是边的尺寸至多为2的超图,极小正则2-图是不含有真正则因子的正则2-图. 设f₂(n)为所有n个顶点的极小正则2-图的最大度数.给出了极小正则2-图的一个结构性质，并由此证得 f₂(n) =(n+3-i)/3, 其中1≤i≤6, n≥7, i≡n(mod 6),从而解决了范红兵等人提出的一个猜想. 作为在图论中的应用, 可以刻画不可分解因子的正则图, 并给出关于度条件的最好可能的因子存在性定理. 进而, f₂(n)和极小2-图可应用于最初引发这项研究的通用开关盒设计问题.     

Z_2Z_4-加性负循环码的对偶

在Z_2Z_4-加性码的基础上研究其循环码,进一步地引入其负循环码.通过建立Z_2Z_4下的正交关系,得出其对偶仍是一个Z_2Z_4负循环码;通过在Z_2Z_4码与Z_4[x]-子模之间建立同构映射来刻画其负循环码的结构以及码的参数类型,并用构造性的方法推出了其对偶的最小生成集.这些结果,便于码元等参数的计算及其应用.     

图的最小特征值

In this paper we investigate the least eigenvalue of a graph whose complement is connected, and present a lower bound for the least eigenvalue of such graph. We also characterize the unique graph whose least eigenvalue attains the second minimum among all graphs of fixed order.     

On perfectly two-edge connected graphs

This paper studies the graphs for which the 2-edge connected spanning subgraph polytope is completely described by the trivial inequalities and the so-called cut inequalities. These graphs are called perfectly 2-edge connected. The class of perfectly 2-edge connected graphs contains for instance the class of series-parallel graphs. We introduce a new class of perfectly 2-edge connected graphs. We discuss some structural properties of graphs which are (minimally with respect to some reduction operations) nonperfectly 2-edge connected. Using this we give sufficient conditions for a graph to be perfectly 2-edge connected.     

The least eigenvalue of a graph with cut vertices

In this paper we characterize the unique graph whose least eigenvalue attains the minimum among all connected graphs of fixed order and given number of cut vertices, and then obtain a lower bound for the least eigenvalue of a connected graph in terms of the number of cut vertices. During the discussion we also get some results for the spectral radius of a connected bipartite graph and its upper bound.     

Minimal least eigenvalue of connected graphs of order n and size m = n + k (5≤k≤8)

The least eigenvalue of a connected graph is the least eigenvalue of its adjacency matrix. We characterize the connected graphs of order n and size n + k (5≤k≤8 and n>k + 5) with the minimal least eigenvalue.     

Total domination in 2‐connected graphs and in graphs with no induced 6‐cycles

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ_t(G) of G. It is known [J Graph Theory 35 (2000), 21–45] that if G is a connected graph of order n > 10 with minimum degree at least 2, then γ_t(G) ≤ 4n/7 and the (infinite family of) graphs of large order that achieve equality in this bound are characterized. In this article, we improve this upper bound of 4n/7 for 2‐connected graphs, as well as for connected graphs with no induced 6‐cycle. We prove that if G is a 2‐connected graph of order n > 18, then γ_t(G) ≤ 6n/11. Our proof is an interplay between graph theory and transversals in hypergraphs. We also prove that if G is a connected graph of order n > 18 with minimum degree at least 2 and no induced 6‐cycle, then γ_t(G) ≤ 6n/11. Both bounds are shown to be sharp. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 55–79, 2009     

Bicyclic graphs for which the least eigenvalue is minimum

The spread of a graph is defined to be the difference between the greatest eigenvalue and the least eigenvalue of the adjacency matrix of the graph. In this paper we determine the unique graph with minimum least eigenvalue among all connected bicyclic graphs of order n. Also, we determine the unique graph with maximum spread in this class for each n?28.     

On the second largest Laplacian eigenvalues of graphs

The second largest Laplacian eigenvalue of a graph is the second largest eigenvalue of the associated Laplacian matrix. In this paper, we study extremal graphs for the extremal values of the second largest Laplacian eigenvalue and the Laplacian separator of a connected graph, respectively. All simple connected graphs with second largest Laplacian eigenvalue at most 3 are characterized. It is also shown that graphs with second largest Laplacian eigenvalue at most 3 are determined by their Laplacian spectrum. Moreover, the graphs with maximum and the second maximum Laplacian separators among all connected graphs are determined.     

The least eigenvalue of signless Laplacian of graphs under perturbation

We investigate how the least eigenvalue of the signless Laplacian of a graph changes by relocating a bipartite branch from one vertex to another vertex, and minimize the least eigenvalue of the signless Laplacian among the class of connected graphs with fixed order which contains a given non-bipartite graph as an induced subgraph.     

Forbidden subgraphs that imply 2-factors

The connected forbidden subgraphs and pairs of connected forbidden subgraphs that imply a 2-connected graph is hamiltonian have been characterized by Bedrossian [Forbidden subgraph and minimum degree conditions for hamiltonicity, Ph.D. Thesis, Memphis State University, 1991], and extensions of these excluding graphs for general graphs of order at least 10 were proved by Faudree and Gould [Characterizing forbidden pairs for Hamiltonian properties, Discrete Math. 173 (1997) 45-60]. In this paper a complete characterization of connected forbidden subgraphs and pairs of connected forbidden subgraphs that imply a 2-connected graph of order at least 10 has a 2-factor will be proved. In particular it will be shown that the characterization for 2-factors is very similar to that for hamiltonian cycles, except there are seven additional pairs. In the case of graphs of all possible orders, there are four additional forbidden pairs not in the hamiltonian characterization, but a claw is part of each pair.