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Laplacian spread of a graph is the difference among the largest and the second smallest Laplacian eigenvalue of the graph. Is there any result or conjecture that discusses about the graphs having minimum Laplacian spread in the class of all connected graphs with prescribed order $n$.

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The following reference shows that the minimum is only attained for path graphs. (corrolary 4.3):

David A. Gregory, Daniel Hershkowitz, Stephen J. Kirkland, "The spread of the spectrum of a graph", Linear Algebra and its Applications, 332–334 (2001) 23–35, https://doi.org/10.1016/S0024-3795(00)00086-0. (https://www.sciencedirect.com/science/article/pii/S0024379500000860)

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