The Laplacian and normalized Laplacian energy of G are given by expressions EL(G) = ∑n i=1 |μi − d|, EL(G) = ∑n i=1 |λi − 1|, respectively, where μi and λi are the eigenvalues of Laplacian matrix L and normalized Laplacian matrix L of G. An interesting problem in spectral graph theory is to find graphs {L,L}−noncospectral with the same E{L,L}(G). In this paper, we present graphs of order n, whi...

Spectral graph theory is the study of properties of the Laplacian matrix or adjacency matrix associated with a graph. In this paper, we focus on the connection between the eigenvalues of the Laplacian matrix and graph connectivity. Also, we use the adjacency matrix of a graph to count the number of simple paths of length up to 3.

in this paper, we present a computational method for solving boundary integral equations with loga-rithmic singular kernels which occur as reformulations of a boundary value problem for the laplacian equation. themethod is based on the use of the galerkin method with cas wavelets constructed on the unit interval as basis.this approach utilizes the non-uniform gauss-legendre quadrature rule for ...

It is shown that if L and D are the Laplacian matrix and the distance matrix of a tree respectively, then any minor of the Laplacian equals the sum of the cofactors of the complementary submatrix of D, upto a sign and a power of 2. An analogous, more general result is proved for the Laplacian and the resistance matrix of any graph. A similar identity is proved for graphs in which each block is ...

R. B. Bapat 2 A. K. Lal3 Sukanta Pati 4 Abstract We consider a q-analogue of the distance matrix (called the q-distance matrix) of an unweighted tree and give formulae for the inverse and the determinant, which generalize the existing formulae for the distance matrix. We obtain the Smith normal form of the q-distance matrix of a tree. The relationship of the q-distance matrix with the Laplacian...

Since the introduction of LLE (Roweis and Saul, 2000) and Isomap (Tenenbaum et al., 2000), a large number of non-linear dimensionality reduction techniques (manifold learners) have been proposed. Many of these non-linear techniques can be viewed as instantiations of Kernel PCA; they employ a cleverly designed kernel matrix that preserves local data structure in the “feature space” (Bengio et al...

Detecting protein complexes is an important way to discover the relationship between network topological structure and its functional features in protein-protein interaction (PPI) network. The spectral clustering method is a popular approach. However, how to select its optimal Laplacian matrix is still an open problem. Here, we analyzed the performances of three graph Laplacian matrices (unnorm...

The regularization functional induced by the graph Laplacian of a random neighborhood graph based on the data is adaptive in two ways. First it adapts to an underlying manifold structure and second to the density of the data-generating probability measure. We identify in this paper the limit of the regularizer and show uniform convergence over the space of Hölder functions. As an intermediate s...

The Laplacian spread of a graph is defined as the difference between the largest and second smallest eigenvalues of the Laplacian matrix of the graph. In this paper, bounds are obtained for the Laplacian spread of graphs. By the Laplacian spread, several upper bounds of the Nordhaus-Gaddum type of Laplacian eigenvalues are improved. Some operations on Laplacian spread are presented. Connected c...