The spectra of the adjacency matrix and Laplacian matrix for some balanced trees
Oscar Rojo,Ricardo Soto +1 more
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TLDR
In this paper, the eigenvalues of the adjacency matrix and of the Laplacian matrix of an unweighted rooted tree of k levels such that in each level the vertices have equal degree were found.About:
This article is published in Linear Algebra and its Applications.The article was published on 2005-07-01 and is currently open access. It has received 72 citations till now. The article focuses on the topics: Degree matrix & Seidel adjacency matrix.read more
Citations
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On the Aα-spectra of trees
TL;DR: In this article, it was shown that the spectral radius of a tree of maximal degree Δ satisfies the tight inequality ρ ( A α ( T Δ ) ) α Δ + 2 ( 1 − α ) Δ − 1, which implies previous bounds of Godsil, Lovasz, and Stevanovic.
Posted Content
Changepoint Detection over Graphs with the Spectral Scan Statistic
TL;DR: In this paper, the spectral scan statistic is proposed to find the sparsest cut in a graph, and its performance as a testing procedure depends directly on the spectrum of the graph and use this result to explicitly derive its asymptotic properties.
Proceedings Article
Changepoint Detection over Graphs with the Spectral Scan Statistic
TL;DR: In this article, the spectral scan statistic is proposed for change-point detection in graphs, and its performance as a testing procedure depends directly on the spectrum of the graph, and use this result to explicitly derive its asymptotic properties on few graph topologies.
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An explicit formula for eigenvalues of Bethe trees and upper bounds on the largest eigenvalue of any tree
Oscar Rojo,María Robbiano +1 more
TL;DR: In this paper, the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of a Bethe tree is derived in terms of the largest vertex degree and the radius of the tree.
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Locating the Eigenvalues of Trees
David P. Jacobs,Vilmar Trevisan +1 more
TL;DR: In this article, a method based on Sylvester's Law of Inertia has been proposed to compute the nonzero eigenvalues of a caterpillar, i.e. how many eigen values lie within the interval of a tree T and interval (α, β ).
References
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Algebraic connectivity of graphs
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Laplacian matrices of graphs: a survey
TL;DR: In this paper, the authors survey some of the many results known for Laplacian matrices, and present a survey of the most important results in the field of graph analysis.
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Tight bounds on the algebraic connectivity of a balanced binary tree
TL;DR: Quite tight lower and upper bounds are obtained on the algebraic connectivity, namely, the second-smallest eigenvalue of the Laplacian matrix, of an unweighted balanced binary tree with k levels and hence n = 2 1 vertices.