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Sharp lower bounds on the Laplacian eigenvalues of trees

โœ Scribed by Kinkar Ch. Das


Book ID
108198693
Publisher
Elsevier Science
Year
2004
Tongue
English
Weight
262 KB
Volume
384
Category
Article
ISSN
0024-3795

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Sharp bounds on the eigenvalues of trees
โœ Shengbiao Hu ๐Ÿ“‚ Article ๐Ÿ“… 2008 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 289 KB

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

Two sharp upper bounds for the Laplacian
โœ Xiao-Dong Zhang ๐Ÿ“‚ Article ๐Ÿ“… 2004 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 168 KB

In this paper, we first obtain a sharp upper bound for the eigenvalues of the adjacency matrix of the line graph of a graph. Then this result is used to present a sharp upper bound for the Laplacian eigenvalues. Another sharp upper bound is presented also. Moreover, we determine all extreme graphs w

Lower bounds for the eigenvalues of Lapl
โœ Abraham Berman; Xiao-Dong Zhang ๐Ÿ“‚ Article ๐Ÿ“… 2000 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 70 KB

We give a lower bound for the second smallest eigenvalue of Laplacian matrices in terms of the isoperimetric number of weighted graphs. This is used to obtain an upper bound for the real parts of the nonmaximal eigenvalues of irreducible nonnegative matrices.

Sharp bound of the kth eigenvalue of tre
โœ Jiansheng Chen ๐Ÿ“‚ Article ๐Ÿ“… 1994 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 585 KB

The sharp lower bound of the kth largest positive eigenvalue of a tree T with n vertices, and the sharp lower bound of the positive eigenvalues of such a tree Tare worked out in this study. A conjecture on the sharp bound of the kth eigenvalue of such a T is proved.