图的双罗马控制数是图论近年来的热点问题之一.通过最大度条件,用双罗马控制数对图的连通性进行刻画,并给出图的某些结构性质.基于树和块图,给出双罗马控制数与控制数之间等式关系的充要条件.

为增强控制系统稳定性和精度,可以通过图论参数来界定图的双罗马控制数,以解决优化问题.文章首先利用双罗马控制数与控制数之间的关系,描述图中赋值为2的顶点个数满足的条件,并根据树的结构性质,给出树中叶子点和支撑点的一个赋值特点.其次引入图论参数,通过参数的概念及特性,得到双罗马控制数与最大度、生成树、3-彩虹控制有关的下界,同时给出双罗马控制数与最小覆盖数、打包数、意大利控制数有关的上界,建立双罗马控制数与图论参数之间联系,进一步表明双罗马控制在刻画图的性质中发挥着重要作用.

在结构图论中,图的哈密尔顿性的谱刻画是最具有影响力的课题之一,其主要思想是判断一个图是不是哈密尔顿图,这是NP-完全问题.因此,诸多学者对哈密尔顿性问题的研究主要集中在寻找适当的充分条件.本文借助补图的无符号拉普拉斯谱半径来刻画具有较大最小度的图的哈密尔顿性.首先,采用反证法构造了原图的闭包,将原图是否具有某性质转化到其闭包中;其次对闭包补图的结构进行了合理的分类讨论;最后分别给出了具有较大最小度的图G是哈密尔顿的,哈密尔顿-连通的以及从任意点出发可迹的关于无符号拉普拉斯谱半径的充分条件.

图的邻接谱半径和无符号拉普拉斯谱半径是描述图的结构和性质的重要工具.本文从哈密尔顿图的边条件出发,通过分析图的度序列,改进图的哈密尔顿性的边数条件,并在此基础上利用图的谱半径以及无符号拉普拉斯谱半径去刻画图的哈密尔顿性.

判断给定的图是不是哈密尔顿图是一个重要的NP-完全问题.图的谱理论就是研究如何通过一些容易计算的不变量来描述图的性质,它是代数图论和组合矩阵论的一个十分重要的研究领域.本文将Aα-谱半径和图的哈密尔顿性联系在一起,分别给出了具有最小度数条件的连通图是哈密尔顿-连通的、哈密尔顿的、可迹的谱充分条件.研究目的在于推广无符号拉普拉斯谱半径到Aα-谱半径,进而讨论图的哈密尔顿性,以此建立图的拓扑结构.

《安庆师范大学学报(自然科学版)》(以下简称本刊)自1982年创刊以来,一直是高校师生发布教研、科研成果的重要平台,更是年轻教师、硕士研究生步入科研大门的重要阶梯,水平越来越高,影响力越来越大.作为一名在安庆师范学院毕业后一直在这里学习工作的安庆师大铁粉,我随学报成长,学报促我成熟,我爱我们的学报.

Let G beagraphoforder n withadjacencymatrix A ( G ) anddiagonalmatrix D ( G ) .For any real α ∈ [ 0 , 1 ] , denote A α ( G ) := α D ( G ) + ( 1 − α) A ( G ) be A α -matrix of graph G . The eigenvalues of A α ( G ) are λ 1 ( A α ( G )) ≥ λ 2 ( A α ( G )) ≥ · · · ≥ λ n ( A α ( G )) , the largest eigenvalue λ 1 ( A α ( G )) is called the A α -spectral radius of G . The A α - separator S A α ( G ) of graph G is defined as S A α ( G ) = λ 1 ( A α ( G )) − λ 2 ( A α ( G )) . For twodisjointgraphs G 1 and G 2 (where V ( G 1 ) and V ( G 2 ) aredisjointwith v 1 ∈ V ( G 1 ) , v 2 ∈ V ( G 2 ) ); the coalescence of G 1 and G 2 with respect to v 1 and v 2 is formed by identifying v 1 and v 2 and is denoted by G 1 · G 2 . The A α -characteristic polynomial of G is defined to be (cid:4)( A α ; x ) = det ( x I n − A α ( G )) , where I n is the identity matrix of size n . A unicyclic graph is a simple connected graph in which the number of edges is equal to the number of vertices. In this paper, firstly, we give the A α -characteristic polynomial of the coalescent graph, and A α -eigenvalues of the star graph for the application. Secondly, we study the extremal graphs with the maximum and minimum A α -spectral radius of the unicyclic graph. Finally, we present the extremal graph with the maximum A α -separator of the unicyclic graph and calculate the range of A α - separator of the corresponding extremal graph.

图的谱半径和无符号拉普拉斯谱半径在研究图的结构和性质中发挥着重要作用.本文从边条件出发,通过计算得出度序列并刻画其对应的图形,进而对这些满足给定边条件的图是否为哈密尔顿图进行了研究,在此基础上把图的谱半径、无符号拉普拉斯谱半径与边条件联系在一起,对图的哈密尔顿性的充分条件进行了探讨.通过边条件和谱条件探讨图的哈密尔顿性是研究图哈密尔顿性的一种可行、有效的方法.

图的周长问题一直是个前沿问题,图的哈密尔顿性是周长取值的一种特殊情况.通过对图的谱刻画,给出周长为c(c≤n-1)的图的谱半径上界、无符号拉普拉斯谱半径上界及哈密尔顿图的谱充分条件,并通过具体实例求出了一个特殊图的谱半径和无符号拉普拉斯谱半径.

Let G be a graph of order n with adjacency matrix A ( G ) and diagonal matrix D ( G ). For any real α∈ [0,1] , denote A_α(G):=α D(G)+(1-α ) A(G) be A_α -matrix of graph G . The eigenvalues of A_α(G) are λ _1(A_α(G))≥λ _2(A_α(G))≥⋯≥λ _n(A_α(G)) , the largest eigenvalue λ _1(A_α(G)) is called the A_α -spectral radius of G . The A_α -separator S_A_α(G) of graph G is defined as S_A_α(G)=λ _1(A_α(G))-λ _2(A_α(G)) . For two disjoint graphs G_1 and G_2 (where V(G_1) and V(G_2) are disjoint with v_1∈ V(G_1) , v_2∈ V(G_2) ); the coalescence of G_1 and G_2 with respect to v_1 and v_2 is formed by identifying v_1 and v_2 and is denoted by G_1· G_2 . The A_α -characteristic polynomial of G is defined to be Φ (A_α;x)=det(xI_n-A_α(G)) , where I_n is the identity matrix of size n . A unicyclic graph is a simple connected graph in which the number of edges is equal to the number of vertices. In this paper, firstly, we give the A_α -characteristic polynomial of the coalescent graph, and A_α -eigenvalues of the star graph for the application. Secondly, we study the extremal graphs with the maximum and minimum A_α -spectral radius of the unicyclic graph. Finally, we present the extremal graph with the maximum A_α -separator of the unicyclic graph and calculate the range of A_α -separator of the corresponding extremal graph.

本文提出图的主独立数概念,研究图的主独立数的性质特征,主要给出了完全二部图、星等特殊图的主独立数,并讨论了一些图的主独立数的界,及给定主独立数的连通图的最小阶问题和给定度δ的n阶图的主独立数的最大值.

本文引入图的符号星独立函数的概念,给出图的符号星独立数的概念以及与之相关的一些基本结论:图的符号星独立数的上、下界,二部图符号星独立数的下界,单圈图、二部图、欧拉图、完全图的符号星独立数.

如果一个简单图中有一条包含图中所有顶点的路,则称这条路为哈密尔顿路;如果图中任意两点都有哈密顿路相连,则称该图是哈密尔顿-连通图.如何判定一个给定的图是否是哈密尔顿-连通图是图论中一个N-P问题,本文主要利用哈密尔顿-连图的闭包运算、边数充分条件以及补图与原图的边数之间的关系,研究并给出利用图的拉普拉斯谱平方和来判定原图是否是哈密尔顿-连通图的充分条件.

图G的亏损数def(G)是指G中所有顶点个数与它的最大匹配中顶点个数之差,如果def(G)≤β,则称图G是β-亏损的.本文主要利用图的Wiener指数、hyper-Wiener指数、Harary指数,给出了具有最小度条件的连通图是β-亏损的充分条件.

图的拉普拉斯矩阵最大特征值定义为图的拉普拉斯谱半径,它是刻画图结构性质的重要参数.本文主要介绍了在所有给定独立数为α的n阶树中具有最大拉普拉斯谱半径的唯一极图,其中[n/2]≤α≤(n-1).

This paper is concerned with a class of Riemann-Liouville fractional-order competitive neural networks with time-varying delay and different time scales. Based on delay-partitioning approach, we construct two suitable Lyapunov functionals including fractional integral terms, respectively, and avoid computing their fractional-order derivatives to derive the synchronization conditions. The sufficient conditions are proposed to ensure the complete synchronization between fractional-order response system and fractional-order derive system. By solving the algebraic equalities or linear matrix inequalities (LMIs), the design of the gain matrix of the linear feedback controller can be realized. An illustrative example is also presented to show the validity and feasibility of the theoretical results.

This paper focuses on discussing the synchronization stability problem of the Riemann–Liouville fractional coupled complex interconnected delayed neural networks. Several globally asymptotic stability criteria for considered network systems are established in terms of Lyapunov functional approach and linear matrix inequality technique. The relationships between the stability of fractional-order isolated neural networks and the synchronization of fractional-order coupled complex neural networks are discussed in detail. The main advantage is to avoid calculating the fractional derivative of Lyapunov functional to discuss the synchronization stability conditions. A numerical example is also performed to demonstrate the validity and feasibility of the proposed results.

Let Snc be the set of all connected graph each of which is a complement of an n-vertex unicyclic graph. Li and Wang (2012) determined the graphs with the least signless Laplacian eignvalue among all the graphs of the complements of n-vertex trees. In this paper, as a continuance of it, the unique graph among Snc which minimizes the least signless Laplacian eigenvalue is identified.