Let G be a graph. We say that a hypergraph H is Berge-G if there exists a bijection phi from E(G) to E(H) such that e subset of phi(e) for all e is an element of E(G). For any r-uniform hypergraph H and a real number p >= 1, the p-spectral radius lambda((p))(H) of H is defined as max(x is an element of R)n,parallel to x parallel to(p)=1 r Sigma({i1,...,ir}is an element of E(H)) x(i1) & centerdot; & centerdot; & centerdot; x(ir). We study the p-spectral radius of Berge-G hypergraphs and determine the 3-uniform hypergraphs with maximum p-spectral radius for p >= 1 among Berge-G hypergraphs when G is a tree that looks like Y with n vertices.

A nontrivial connected graph $ G $ with diameter $ d $ can be assigned a red-white coloring, where the vertices of $ G $ are colored either red or white, with the stipulation that at least one vertex must be red. Associated with each vertex $ v $ of $ G $ is a $ d $-vector, called the code of $ v $, whose $ i $th coordinate is the number of red vertices at distance $ i $ from $ v $. A red-white coloring of $ G $ for which distinct vertices have distinct codes is called an identification coloring or $ ID $-coloring of $ G $. A graph $ G $ possessing an $ ID $-coloring is called an $ ID $-graph. The minimum number of red vertices among all $ ID $-colorings of an $ ID $-graph $ G $ is the identification number or $ ID $-number of $ G $. The number of red vertices in an identification coloring is called the identification coloring number. This article studied the identification coloring number of lollipop graphs by constructing vertex colorings.

Online education has become a popular teaching method beyond classroom education. However, online teaching resources bring the software and hardware resources cost and security burden of cloud computing, which limits the willingness to share online resources. How to effectively motivate different schools to share teaching resources is a challenging issue. This paper proposes an excitation mechanism to promote high-quality online resource sharing. Firstly, under the resource constraints of the cloud platform, a utility oriented resource sharing model is established. Then, a game of choice between students and course resources is studied, which allows users to participate course strategy selection. Next, a particle swarm optimization algorithm was used to design an optimal cloud computing resource allocation scheme that maximizes revenue. It expects that high-quality course resources will be allocate more cloud computing resources. Finally, experiment simulation shows that appropriately excitation mechanisms can effectively promote the sharing of teaching resources, furthermore, the proposed resource allocation scheme is significantly better than the average allocation scheme.

图的哈密尔顿问题一直以来都是图论研究的重点和难点.由于图的谱和拓扑指数便于计算,近年来人们开始利用其优势来研究图的哈密尔顿性.受此启发,首先根据平衡二部图是弱哈密尔顿-连通的边充分条件得到拟平衡二部图是弱逐点可迹的边充分条件;其次利用图的谱半径及无符号拉普拉斯谱半径分别给出了拟平衡二部图是弱逐点可迹的充分条件;最后利用图的Wiener指数、Hyper-Wiener指数以及Harary指数分别给出了拟平衡二部图是弱逐点可迹的充分条件.

设G是一个n阶k圈图,k圈图为边数等于顶点数加k-1的简单连通图.μ1(G)、μ2(G)分别记为图G的Laplace矩阵的最大特征值和次大特征值,图G的Laplace分离度定义为SL(G)=μ1(G)-μ2(G).本文研究了给定阶数的k圈图的最大Laplace分离度,并刻画了相应的极图,其结果推广了已有当k=1,2,3时的结论.

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.

Let G be a simple graph. The adjacency matrix is denoted by G. The least eigenvalue of A(G), denoted by λmin(G), is called the least eigenvalue of G. In this paper we first establish the relations between the number of edges and the least eigenvalue of the adjacency matrix of the graph, and then give some spectral conditions for a graph having Hamiltonian paths or Hamiltonian cycles, or being Hamilton-connected, or being traceable from every vertex in terms of the least eigenvalue of the adjacency matrix of the graph. This provides an effective method for us to study some properties for a graph.

Let G be a graph, and the number of components of G is denoted by c(G). Let t be a positive real number. A connected graph G is t-tough if tc(G − S) ≤ |S| for every vertex cut S of V(G). The toughness of G is the largest value of t for which G is t-tough, denoted by τ(G). We call a graph G Hamiltonian if it has a cycle that contains all vertices of G. Chvátal and other scholars investigate the relationship between toughness conditions and the existence of cyclic structures. In this paper, we establish some sufficient conditions that a graph with toughness is Hamiltonian based on the number of edges, spectral radius, and signless Laplacian spectral radius of the graph.MR subject classifications: 05C50, 15A18.

由于图的谱能够很好地反映图的结构性质且便于计算,因而可以利用图谱理论来研究图的哈密尔顿性.主要研究哈密尔顿图的谱充分条件和无符号拉普拉斯谱充分条件.首先介绍图的闭包性质;然后对图的闭包结构进行分析、论证,利用度序列以及反证法找出带有最小度的图是哈密尔顿图的边数充分条件;最后根据图的边数与谱半径、无符号拉普拉斯谱半径的关系,分别给出G是哈密尔顿图的谱充分条件、无符号拉普拉斯谱充分条件.所得到的结论均优化已有结论.

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.

A pancyclic graph of order n is a graph with cycles of all possible lengths from 3 to n. In fact, it is NP-complete that deciding whether a graph is pancyclic. Because the spectrum of graphs is convenient to be calculated, in this study, we try to use the spectral theory of graphs to study this problem and give some sufficient conditions for a graph to be pancyclic in terms of the spectral radius and the signless Laplacian spectral radius of the graph.

利用无符号拉普拉斯谱半径与特征向量之间的关系式,研究有n个顶点、最小度为δ且边连通度k′

图的邻接矩阵和无符号拉普拉斯矩阵的最大特征值分别称为谱半径和无符号拉普拉斯谱半径.由于图的谱容易计算,所以通过图的谱来研究图的结构性质.近年来,利用图的谱研究图的哈密尔顿性已经成为一个前沿热点问题.受此启发,利用图以及补图的谱半径和无符号拉普拉斯谱半径来刻画图的哈密尔顿性,进而得到更好的哈密尔顿图的谱充分条件.

A path passing through all the vertices of a graph is called a Hamilton path .The graph G is said to be Hamilton-connected if any two vertices of G are connected by a Hamilton path .The graph G is traceable from any vertex if it contains a Hamilton path from every vertex of G .In terms of the edge number , the spectral radius and the signless Laplacian spectral radius of a graph ,some sufficient conditions for the graph to be Hamilton-connected and to be traceable from every vertex were presented ,respectively .

本文研究并给出了禁用kP3的谱必要条件.以禁用kP3的边数必要条件为出发点,考虑到图的极端谱与边数之间的联系,利用图G的邻接谱半径给出了禁用kP3的谱必要条件,并利用图G的无符号拉普拉斯谱半径给出了禁用kP3的无符号拉普拉斯谱必要条件,同时证明了相应的定理.

Let G=(V (G) ,E(G)) be a simple graph of order n and size m .The inverse degree ofa graph G with no isolated vertices is defined by ID(G)= ∑vi ∈ V(G)1/ d(vi) ,where d(vi)is the degree1 of the vertex v i ∈ V (G ) . First , in terms of the inverse degree , sufficient conditions for a connected graph to be k-Hamiltonian , k-edge-Hamiltonian , k-path-coverable , Hamilton-connected , k-connected ,2-edge-connected and β-deficient were obtained ,respectively .Second , sufficient conditions for the independence number of a connected graph to be less than or equal to the integer k were given .Finally ,a sufficient condition for a connected balanced bipartite graph to be Hamiltonian was given .

The reciprocal degree resistance distance index of a connected graph G is defined as RDR G = ∑ u , v ⊆ V G d G u + d G v / r G u , v , where r G u , v is the resistance distance between vertices u and v in G . Let ℬ n denote the set of bicyclic graphs without common edges and with n vertices. We study the graph with the maximum reciprocal degree resistance distance index among all graphs in ℬ n and characterize the corresponding extremal graph.

Let $G(V,E)$ be a simple connected graph of order $n$. A graph of order $n$ is called pancyclic if it contains all the cycles $C_k$ for $k\in \{3,4,\cdot\cdot\cdot,n\}$. In this paper, some new spectral sufficient conditions for the graph to be pancyclic are established in terms of the edge number, the spectral radius and the signless Laplacian spectral radius of the graph.