We provide an analysis of the expected meeting time of two independent random walks on a regular graph. For 1-D circle and 2-D torus graphs, we show that the expected meeting time can be expressed as the sum of the inverse of non-zero eigenvalues of a suitably defined Laplacian matrix. We also conjecture based on empirical evidence that this result holds more generally for simple random walks o...

We consider a canonical ensemble of synchronization-optimized networks of identical oscillators under external noise. By performing a Markov chain Monte Carlo simulation using the Kirchhoff index, i.e., the sum of the inverse eigenvalues of the Laplacian matrix (as a graph Hamiltonian of the network), we construct more than 1,000 different synchronization-optimized networks. We then show that t...

A classical matrix-tree theorem expresses the determinant of some matrix constructed from a graph (principal minor of the Laplacian) as a sum over all spanning trees of the graph. There are generalizations of this theorem to hypergraphs or simplicial complexes [MV, DKM]. Some version of this theorem provides a formula for the first non-zero coefficient of the Conway polynomial of a (virtual) li...

A general two-layer network consists of two networks G_{1} and G_{2}, whose interconnection pattern is specified by the interconnectivity matrix B. We deduce desirable properties of B from a dynamic process point of view. Many dynamic processes are described by the Laplacian matrix Q. A regular topological structure of the interconnectivity matrix B (constant row and column sum) enables the com...

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 a tight lower bound for algebraic connectivity of graphs (second smallest eigenvalue of the Laplacian matrix of the graph) based on connection-graph-stability method is introduced. The connection-graph-stability score for each edge is defined as the sum of the length of all the shortest paths making use of that edge. We prove that the algebraic connectivity of the graph is lower b...

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 ...

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...