Abstract
In this paper, we first give an upper bound for the largest signless Laplacian eigenvalue of a graph and find all the extremal graphs. Secondly, we consider the second-largest signless Laplacian eigenvalue and we characterize the connected graphs whose second-largest signless Laplacian eigenvalue does not exceed 3. Furthermore, we give the signless Laplacian spectral characterization of the latter graphs. In particular, the well-known friendship graph is proved to be determined by the signless Laplacian spectrum.
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