On the sth Laplacian eigenvalue of trees of order st+1 Original Research Article

Let image be the Laplacian eigenvalues of a simple undirected graph image. Let sgreater-or-equal, slanted2 and tgreater-or-equal, slanted2 be integers and let image be the rooted tree of three levels and order st+1 such that the vertex root has degree s, the vertices in level 2 have degree t and the s(t-1) pendants vertices are in level 3. We prove thatimageThis result solves a conjecture due to Shao et al. in [J.-y. Shao, L. Zhang, X.-y. Yuan, On the second Laplacian eigenvalue of trees of odd order, Linear Algebra Appl. 419 (2006) 475–485].