Abstract
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 ≥ 4, where K n − e is the n vertex complete graph with an edge removed. Moreover, examples of such graphs are given in this article.
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Acknowledgments
The authors gratefully acknowledge valuable suggestions from the referee who helped to considerably improve this article. Research by G.R. Omidi was in part supported by a grant from IPM (No. 88050012).