Laplacian eigenvalue distribution, diameter and domination number of trees

For a graph G with domination number γ, Hedetniemi, Jacobs and Trevisan (2016) proved that mG[0,1)≤γ, where mG[0,1) means the number of Laplacian eigenvalues of G in the interval [0,1). Let T be a tree with diameter d. In this paper, we show that 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 if and only if it has exactly (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.