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...

We study details of geometry noncommutative de Sitter space: we determine the Riemann and Ricci curvature tensors, energy Laplacian. find, in particular, that fuzzy space is an Einstein space, $R_{ab}=-3\zeta\,\eta_{ab}$. The Laplacian, defined frame formalism, not hermitian gives nonunitary evolution. When symmetrically ordered, it has usual quadratic form $\Delta=\Pi_a\Pi^a$ (when acting on f...

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...

Recently, Braunstein et al. [1] introduced normalized Laplacian matrices of graphs as density matrices in quantum mechanics and studied the relationships between quantum physical properties and graph theoretical properties of the underlying graphs. We provide further results on the multipartite separability of Laplacian matrices of graphs. In particular, we identify complete bipartite graphs wh...

In this paper, bi-matrix games are investigated based on L-R fuzzy variables. Also, based on the fuzzy max order several models in non-symmetrical L-R fuzzy environment is constructed and the existence condition of Nash equilibrium strategies of the fuzzy bi-matrix games is proposed. At last, based on the Nash equilibrium of crisp parametric bi-matrix games, we obtain the Pareto and weak Pareto...

In this paper, we consider the Lévy Laplacian acting on multiple Wiener integrals by the Lévy process, and give a necessary and sufficient condition for eigenfunctions of the Lévy Laplacian. Moreover we give a decomposition by eigenspaces consisting of multiple Wiener integrals by the Lévy process in terms of the Lévy Laplacian.

Let M denote the Laplacian matrix of a graph G. Associated with G is a finite group Φ(G), obtained from the Smith normal form of M , and whose order is the number of spanning trees of G. We provide some general results on the relationship between the eigenvalues of M and the structure of Φ(G), and address the question of how often the group Φ(G) is cyclic.