Let B be a weighted generalized Bethe tree of k levels (k > 1) in which nj is the number of vertices at the level k− j+1 (1 ≤ j ≤ k). Let ∆ ⊆ {1, 2, . . . , k − 1} and F= {Gj : j ∈ ∆}, where Gj is a prescribed weighted graph on each set of children of B at the level k−j+1. In this paper, the eigenvalues of a block symmetric tridiagonal matrix of order n1 +n2 + · · ·+nk are characterized as the ...

In this paper we present a computationally efficient algorithm utilizing a fully or seminonlocal graph Laplacian for solving a wide range of learning problems in binary data classification and image processing. In their recent work [Multiscale Model. Simul., 10 (2012), pp. 1090–1118], Bertozzi and Flenner introduced a graph-based diffuse interface model utilizing the Ginzburg–Landau functional ...

In this paper, we investigate the use of heat kernels as a means of embedding graphs in a pattern space. We commence by performing the spectral decomposition on the graph Laplacian. The heat kernel of the graph is found by exponentiating the resulting eigensystem over time. By equating the spectral heat kernel and its Gaussian form we are able to approximate the geodesic distance between nodes ...

For a graph G, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of arbitrary dimension. This requires generalizing the notion of rooted forest to higher dimension. We also introduce orientations of higher dimensional rooted trees a...

The m-connectivity index χα(G) of an organic molecule whose molecular graph is G is the sum of the weights (di1di2 ...dim+1) , where i1−i2− ...−im+1 runs over all paths of length m in G and di denotes the degree of vertex vi. We find upper bounds for χα(G) when m ≥ 1 and α ≥ −1 (α 6= 0) using the eigenvalues of the Laplacian matrix of an associated weighted graph.

For a graph G and a real α / = 0, we study the graph invariant sα(G) – the sum of the αth power of the non-zero Laplacian eigenvalues of G. The cases α = 2, 1 2 and −1 have appeared in different problems. Here we establish some properties for sα with α / = 0, 1. We also discuss the cases α = 2, 1 2 . © 2008 Elsevier Inc. All rights reserved. AMS classification: 05C50; 05C90

So far, we have been adopting the usual approach to spectral graph theory: understand graphs via the eigenvectors and eigenvalues of associated matrices. For example, given a graph G = (V,E), we defined an adjacency matrix A and considered the eigensystem Av = λv, and we also defined the Laplacian matrix L = D−A and considered the Laplacian quadratic form xTLx− ∑ (ij)∈E(xi−xj). There are other ...