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On trees with exactly one characteristic element
- Source :
-
Linear Algebra & its Applications . Mar2007, Vol. 421 Issue 2/3, p233-242. 10p. - Publication Year :
- 2007
-
Abstract
- Abstract: Let T be a tree and L be its Laplacian matrix. With respect to a given vector Y which gives a valuation of vertices of T, a vertex u of T is called a characteristic vertex if Y[u]=0 and if there is a vertex w adjacent to u with Y[w]≠0; and an edge {u, w} of T is called a characteristic edge if Y[u]Y[w]<0. The characteristic set of T with respect to Y, denoted by , is defined as the collection of all characteristic vertices and characteristic edges of T. For the kth smallest eigenvalue λ k (k ⩾2) of T, if λk > λ k−1, the corresponding eigenvector Y of λ k is called a k-vector. A tree T is called k-simple if for all k-vectors Y. We show that k-simple trees exist and characterize them. We also show the characteristic sets determined by all the k-vectors is the same, which is consistent with the property of 2-simple tree (i.e., arbitrary tree). Finally, we give some properties of the eigenvalues and eigenvectors of a k-simple tree. [Copyright &y& Elsevier]
- Subjects :
- *LAPLACIAN operator
*PARTIAL differential equations
*EIGENVALUES
*EIGENVECTORS
Subjects
Details
- Language :
- English
- ISSN :
- 00243795
- Volume :
- 421
- Issue :
- 2/3
- Database :
- Academic Search Index
- Journal :
- Linear Algebra & its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 23742788
- Full Text :
- https://doi.org/10.1016/j.laa.2006.04.016