设G是一个n阶图而H是任意一个图.符号G?H表示由G和n个顶点不交的图H通过把G的第i个顶点和第i个H的所有顶点都连一条边所得的图,其中1≤i≤n.设p≥3和q为两个正整数.令Cp和Kp分别表示p个顶点的圈和完全图.证明了Cp?qK_1和Kp?qK_1分别被...设G是一个n阶图而H是任意一个图.符号G?H表示由G和n个顶点不交的图H通过把G的第i个顶点和第i个H的所有顶点都连一条边所得的图,其中1≤i≤n.设p≥3和q为两个正整数.令Cp和Kp分别表示p个顶点的圈和完全图.证明了Cp?qK_1和Kp?qK_1分别被它们的拉普拉斯图谱所确定,且当p为奇数时Cp?qK_1也被它的无符号拉普拉斯图谱所确定.文中的结果推广了[Bu Changjiang, et al.,(2014),Graphs Combin, 30:1123-1133],[Boulet R (2009). Discrete Math Theor Comput Sci, 11:149-160]和[Mirzakhah M, Kiani D (2010). Electron J Linear Algebra, 20:610-620]的相应结论.展开更多
Eigenvectors and eigenvalues of discrete Laplacians are often used for manifold learning and nonlinear dimensionality reduction.Graph Laplacian is one widely used discrete laplacian on point cloud.It was previously pr...Eigenvectors and eigenvalues of discrete Laplacians are often used for manifold learning and nonlinear dimensionality reduction.Graph Laplacian is one widely used discrete laplacian on point cloud.It was previously proved by Belkin and Niyogithat the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled independently from the uniform distribution over the manifold.Recently,we introduced Point Integral method(PIM)to solve elliptic equations and corresponding eigenvalue problem on point clouds.In this paper,we prove that the eigenvectors and eigenvalues obtained by PIM converge in the limit of infinitely many random samples.Moreover,estimation of the convergence rate is also given.展开更多
The small-world network, proposed by Watts and Strogatz, has been extensively studied for the past over ten years. In this paper, a generalized smMl-world network is proposed, which extends severM small-world network ...The small-world network, proposed by Watts and Strogatz, has been extensively studied for the past over ten years. In this paper, a generalized smMl-world network is proposed, which extends severM small-world network models. Furthermore, some properties of a special type of generalized small-world network with given expectation of edge numbers have been investigated, such as the degree distribution and the isoperimetric number. These results are used to present a lower and an upper bounds for the clustering coefficient and the diameter of the given edge number expectation generalized small-world network, respectively. In other words, we prove mathematically that the given edge number expectation generalized small-world network possesses large clustering coefficient and small diameter.展开更多
文摘设G是一个n阶图而H是任意一个图.符号G?H表示由G和n个顶点不交的图H通过把G的第i个顶点和第i个H的所有顶点都连一条边所得的图,其中1≤i≤n.设p≥3和q为两个正整数.令Cp和Kp分别表示p个顶点的圈和完全图.证明了Cp?qK_1和Kp?qK_1分别被它们的拉普拉斯图谱所确定,且当p为奇数时Cp?qK_1也被它的无符号拉普拉斯图谱所确定.文中的结果推广了[Bu Changjiang, et al.,(2014),Graphs Combin, 30:1123-1133],[Boulet R (2009). Discrete Math Theor Comput Sci, 11:149-160]和[Mirzakhah M, Kiani D (2010). Electron J Linear Algebra, 20:610-620]的相应结论.
基金This research was supported by NSFC Grant 11671005.
文摘Eigenvectors and eigenvalues of discrete Laplacians are often used for manifold learning and nonlinear dimensionality reduction.Graph Laplacian is one widely used discrete laplacian on point cloud.It was previously proved by Belkin and Niyogithat the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled independently from the uniform distribution over the manifold.Recently,we introduced Point Integral method(PIM)to solve elliptic equations and corresponding eigenvalue problem on point clouds.In this paper,we prove that the eigenvectors and eigenvalues obtained by PIM converge in the limit of infinitely many random samples.Moreover,estimation of the convergence rate is also given.
基金Supported by National Natural Science Foundation of China(Grant Nos.10971137and11271256)NationalBasic Research Program of China973Program(Grant No.2006CB805900)the Grant of Science andTechnology Commission of Shanghai Municipality(STCSM No.09XD1402500)
文摘The small-world network, proposed by Watts and Strogatz, has been extensively studied for the past over ten years. In this paper, a generalized smMl-world network is proposed, which extends severM small-world network models. Furthermore, some properties of a special type of generalized small-world network with given expectation of edge numbers have been investigated, such as the degree distribution and the isoperimetric number. These results are used to present a lower and an upper bounds for the clustering coefficient and the diameter of the given edge number expectation generalized small-world network, respectively. In other words, we prove mathematically that the given edge number expectation generalized small-world network possesses large clustering coefficient and small diameter.
