Laplacian eigenvalue distribution, diameter and domination number of trees

For a graph G with domination number gamma, Hedetniemi, Jacobs and Trevisan (2016) proved that $ m_{G}[0,1)\leq \gamma $ mG[0,1)<=gamma, where $ m_{G}[0,1) $ mG[0,1) means the number of Laplacian eigenvalues of G in the interval $ [0,1) $ [0,1). Let T be a tree with diameter d. In this paper, we show that $ m_{T}[0,1)\geq (d+1)/3 $ mT[0,1)>=(d+1)/3. All trees achieving the lower bound are completely characterized. Moreover, we prove that the domination number of a tree is $ (d+1)/3 $ (d+1)/3 if and only if it has exactly $ (d+1)/3 $ (d+1)/3 Laplacian eigenvalues less than one. As an application, it also provides a new type of tree, which shows the sharpness of the inequality due to Hedetniemi, Jacobs and Trevisan.