Sharp lower bounds on the Laplacian eigenvalues of trees
โ 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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๐ SIMILAR VOLUMES
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
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
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.
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.