On graphs whose Laplacian matrices have distinct integer eigenvalues
✍ Scribed by Shaun M. Fallat; Stephen J. Kirkland; Jason J. Molitierno; M. Neumann
- Publisher
- John Wiley and Sons
- Year
- 2005
- Tongue
- English
- Weight
- 120 KB
- Volume
- 50
- Category
- Article
- ISSN
- 0364-9024
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✦ Synopsis
Abstract
In this paper, we investigate graphs for which the corresponding Laplacian matrix has distinct integer eigenvalues. We define the set S~i,n~ to be the set of all integers from 0 to n, excluding i. If there exists a graph whose Laplacian matrix has this set as its eigenvalues, we say that this set is Laplacian realizable. We investigate the sets S~i,n~ that are Laplacian realizable, and the structures of the graphs whose Laplacian matrix has such a set as its eigenvalues. We characterize those i < n such that S~i,n~ is Laplacian realizable, and show that for certain values of i, the set S~i,n~ is realized by a unique graph. Finally, we conjecture that S~n,n~ is not Laplacian realizable for n ≥ 2 and show that the conjecture holds for certain values of n. © 2005 Wiley Periodicals, Inc. J Graph Theory