Topic
Spectral graph theory
About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.
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01 Jan 2013
TL;DR: In this article, a generalization of the determinant of the Laplacian matrix of a weighted directed graph is presented, which is a generalisation of Bapat et al. (1998).
Abstract: The notion of weighted directed graph is a generalization of mixed graphs. In this article a formula for the determinant of the Laplacian matrix of a weighted directed graph is obtained. It is a generalization of the formula for the determinant of the Laplacian matrix of a mixed graph obtained by Bapat et al. (Linear Multilinear Algebra 46:299–312, 1999).
4 citations
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TL;DR: This paper generalizes the following lower bound on the chromatic number due to Alan Hoffman to include all eigenvalues of the adjacency, Laplacian and signless LaPLacian matrices.
Abstract: One of the best known results in spectral graph theory is the following lower bound on the chromatic number due to Alan Hoffman, where mu_1 and mu_n are respectively the maximum and minimum eigenvalues of the adjacency matrix: chi >= 1 + mu_1 / (- mu_n). We recently generalised this bound to include all eigenvalues of the adjacency matrix.
In this paper, we further generalize these results to include all eigenvalues of the adjacency, Laplacian and signless Laplacian matrices. The various known bounds are also unified by considering the normalized adjacency matrix, and examples are cited for which the new bounds outperform known bounds.
4 citations
01 Mar 2017
TL;DR: It is shown that for moderate block sizes, certain types of grid symmetry enable us to design non-separable block transforms that have computational complexities comparable to those of separable ones.
Abstract: When a line graph is symmetric, the associated graph Fourier transform has a fast implementation. In this paper, we extend this idea to the 2D non-separable case, where the graph of interest is a square-shaped grid. We investigate a number of symmetry types for 2D grids. Then, for each type of symmetry we derive a block-diagonalization form of the graph Laplacian matrix, based on which fast implementations with reduced number of multiplications can be obtained. We show that for moderate block sizes, certain types of grid symmetry enable us to design non-separable block transforms that have computational complexities comparable to those of separable ones.
4 citations
27 Oct 2008
TL;DR: This work proposes cascading decomposition algorithm based on the spectral analysis on a normalized graph Laplacian to decompose the problem into several sub-problems and conduct parameter relevance analysis on each sub-problem to perform dynamic state abstraction.
Abstract: Problem decomposition and state abstractions applied in the hierarchical problem solving often requires manual construction of a hierarchy structure in advance. This work is to provide some automatic algorithms for dimension reduction in problem solving. We propose cascading decomposition algorithm based on the spectral analysis on a normalized graph Laplacian to decompose the problem into several sub-problems and conduct parameter relevance analysis on each sub-problem to perform dynamic state abstraction. In each decomposed sub-problem, only parameters in the projected state space related to its sub-goal are reserved, and identical sub-problems are integrated into one through feature comparison. The whole problem is transformed into a combination of projected sub-problems, and problem solving in the abstracted space is more efficient. The paper demonstrates the performance improvement on reinforcement learning based on the proposed state space decomposition and abstraction methods using a capture-the-flag scenario.
4 citations
TL;DR: A Cheeger inequality of distance regular graphs is given in terms of the smallest positive eigenvalue of the Laplacian and a value α d which is defined using q -numbers to approximate α d with arbitrarily small positive error β.
4 citations