#### Title

### On Graphs whose Laplacian Matrices have Distinct Integer Eigenvalues

#### Document Type

Peer-Reviewed Article

#### Publication Date

10-2005

#### 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*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}*i*<

*n*such that

*S*is Laplacian realizable, and show that for certain values of

_{i,n}*i*, the set

*S*is realized by a unique graph. Finally, we conjecture that

_{i,n}*S*is not Laplacian realizable for

_{n,n}*n*≥ 2 and show that the conjecture holds for certain values of

*n*.

#### DOI

10.1002/jgt.20102

#### Recommended Citation

Fallat, S., Kirkland, S., Molitierno, J., & Neumann, M. (2005). On graphs whose Laplacian matrices have distinct integer eigenvalues. *Journal of Graph Theory, 50*(2), 162–174. doi: 10.1002/jgt.20102