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The kth Laplacian eigenvalue of a tree

✍ Scribed by Ji-Ming Guo

Publisher
John Wiley and Sons
Year
2006
Tongue
English
Weight
102 KB
Volume
54
Category
Article
ISSN
0364-9024

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✦ Synopsis

Abstract

Let Ξ»~k~(G) be the __k__th Laplacian eigenvalue of a graph G. It is shown that a tree T with n vertices has $\lambda_{k}(T)\le \lceil { {n}\over{k}}\rceil$ and that equality holds if and only if k < n, k|n and T is spanned by k vertex disjoint copies of ${K}_{{1}, { {n}\over {k}}-1}$, the star on ${{n}\over{k}}$ vertices. Β© 2006 Wiley Periodicals, Inc. J Graph Theory

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## Abstract It is shown that there exist domains Ξ© βŠ‚ ℝ^__N__^, which outside of some ball coincide with the strip ℝ^__N__ βˆ’ 1^ Γ— (0, Ο€) and for which the Dirichlet Laplacian – Ξ” has eigenvalues within the subinterval (1, 4) of the essential spectrum (1, ∞).