On Graphs whose Laplacian Matrices have Distinct Integer Eigenvalues

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Peer-Reviewed Article

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In this paper, we investigate graphs for which the corresponding Laplacian matrix has distinct integer eigenvalues. We define the set Si,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 Si,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 Si,n is Laplacian realizable, and show that for certain values of i, the set Si,n is realized by a unique graph. Finally, we conjecture that Sn,n is not Laplacian realizable for n ≥ 2 and show that the conjecture holds for certain values of n.