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Locating the Eigenvalues of Trees

โœ Scribed by David P. Jacobs; Vilmar Trevisan


Publisher
Elsevier Science
Year
2011
Tongue
English
Weight
497 KB
Volume
434
Category
Article
ISSN
0024-3795

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Sharp bounds on the eigenvalues of trees
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Let ฮป 1 (T ) and ฮป 2 (T ) be the largest and the second largest eigenvalues of a tree T , respectively. We obtain the following sharp lower bound for ฮป 1 (T ): where d i is the degree of the vertex v i and m i is the average degree of the adjacent vertices of v i . Equality holds if and only if T i

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The kth Laplacian eigenvalue of a tree
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## Abstract Let ฮป~__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 disjoin