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We investigate graphs whose signless Laplacian matrix has
three distinct eigenvalues. We show that the largest signless
Laplacian eigenvalue of a connected graph $G$ with three distinct
signless Laplacian eigenvalues is noninteger if and only if
$G=K_n-e$ for $n \geq 4$, where $K_n-e$ is the $n$ vertex
complete
graph with an edge removed. Moreover, examples of such graphs
are given.
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