On trees with exactly one characteristic element
type: Item
instance of: scholarly article
title: On trees with exactly one characteristic element
zbMATH Open document ID: 1115.05055
author: Yizheng Fan
author: Shi-Cai Gong
author: Yi Wang
author: Yubin Gao
published in: Linear Algebra and its Applications
publication date: 2007-03-09T00:00:00Z
review text: \loadeusb For a given tree \(T\), its Laplacian matrix \(L\) and a given vector \(Y\) (a valuation of vertices of \(T\)), a vertex \(u\) of \(T\) is called a characteristic vertex if and only if \(Y[u]=0\) and there is a vertex \(w\) adjacent to \(u\) with \(Y[w]\neq 0\); and an edge \(\{u,w\}\) is called a characteristic edge if \(Y[u]Y[w]<0\). The characteristic set of \(T\) with respect to \(Y\), \({\mathcal S}(T,Y)\), is a collection of all characteristic vertices and characteristic edges of \(T\). Let \(\lambda_k\) be the \(k\)th smallest eigenvalue. If \(\lambda_k>\lambda_{k-1}\), the corresponding eigenvector \(Y\) of \(\lambda_k\) is called a \(k\)-vector and a tree \(T\) is called \(k\)- simple if and only if \(| {\mathcal S}(T,Y)| =1\) for all \(k\)-vectors \(Y\). The authors show the existence of \(k\)-simple trees and characterize them. They also show that the characteristic sets determined by all the \(k\)-vectors are the same and give some properties of the eigenvalues and eigenvectors of a \(k\)-simple tree.
reviewed by: Vladimír Lacko
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 5132593
zbMATH Keywords: Laplacian matrix
zbMATH Keywords: eigenvalue
zbMATH Keywords: eigenvector
MaRDI profile type: Publication
full work available at URL: j.laa.2006.04.016
OpenAlex ID: W2060786369
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DOI: 10.1016/J.LAA.2006.04.016
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description: scientific article; zbMATH DE number 5132593