Let G = (V,E) be a graph with vertex set V = {1, . . . , n} and edge set E. Throughout the paper, a graph G is undirected and simple (i.e., has no multi-edges or loops). We allow G to be disconnected. The Laplacian of G is the matrix L(G) = D−A, where D is the diagonal matrix whose entries are the degree of the vertices and A is the adjacency matrix of G. Chung’s normalized Laplacian L̃(G) [6] i...

Let Gl be the graph obtained from Kl by adhering the root of isomorphic trees T to every vertex of Kl, and dk−j+1 be the degree of vertices in the level j. In this paper we study the spectrum of the adjacency matrix A(Gl) and the Laplacian matrix L(Gl) for all positive integer l, and give some results about the spectrum of the adjacency matrix A(Gl) and the Laplacian matrix L(Gl). By using thes...

Let G be a connected graph with vertex set V ( ). The resistance distance between any two vertices u , v ∈ ) is the net effective them in electric network constructed from by replacing each edge unit resistor. S ⊂ of such that all have same neighborhood − and let [ ] subgraph induced . In this note, {1}-inverse Laplacian matrix formula for distances obtained. It turns out could given terms elem...

A new distance measure between probability density functions (pdfs) is introduced, which we refer to as the Laplacian pdf distance. The Laplacian pdf distance exhibits a remarkable connection to Mercer kernel based learning theory via the Parzen window technique for density estimation. In a kernel feature space defined by the eigenspectrum of the Laplacian data matrix, this pdf distance is show...

The understanding of the relationship between topology and behaviour in interconnected networks would allow to charac- terise and predict behaviour in many real complex networks since both are usually not simultaneously known. Most previous studies have focused on the relationship between topology and synchronisation. In this work, we provide analytical formulas that shows how topology drives c...

Let G be a weighted graph with Laplacian matrix L and signless Laplacian matrix Q. In this note, block representations for the group inverse of L and Q are given. The resistance distance in a graph can be obtained from the block representation of the group inverse of L.

Recall that, given a graph G, the matrix Q = D + A is called the signless Laplacian, where A is the adjacency matrix and D is the diagonal matrix of vertex degrees. The matrix L = D − A is known as the Laplacian of G. Graphs with the same spectrum of an associated matrix M are called cospectral graphs with respect to M , or M–cospectral graphs. A graph H cospectral with a graph G, but not isomo...

We consider the p–Laplacian operator on a domain equipped with a Finsler metric. After deriving and recalling relevant properties of its first eigenfunction for p > 1, we investigate the limit problem as p → 1.

Given a C-function f on a compact riemannian manifold (X, g) we give a set of frequencies L = Lf (ε) depending on a small parameter ε > 0 such that the relative L-error ‖f−f ‖ ‖f‖ is bounded above by ε, where f L denotes the L-partial sum of the Fourier series f with respect to an orthonormal basis of L(X) constituted by eigenfunctions of the Laplacian operator ∆ associated to the metric g.

We establish sharp regularity and Fredholm theorems for the ∂̄b-Neumann problem on domains satisfying some non-generic geometric conditions. We use these domains to construct explicit examples of bad behaviour of the Kohn Laplacian: it is not always hypoelliptic up to the boundary, its partial inverse is not compact and it is not globally subelliptic.