On the two conjectures of Graffiti Original Research Article

We study the Laplacian eigenvalues of trees on n vertices with independence number α and describe all extremal graphs that attain the maximal Laplacian spectral radius and algebraic connectivity. Then the results are used to confirm two conjectures of Graffiti (WOW Conjectures 584 and 636) on the relationship between the Laplacian eigenvalues and the independence number of a graph.