Sharp bounds and monotonicity results for Neumann eigenvalues

Abstract: In this article, we study sharp bounds for the Neumann eigenvalues of the Laplace operator on graphs. We first obtain monotonicity results for the Neumann eigenvalues on trees. In particular, we show that increasing any number of boundary vertices while keeping interior vertices unchanged in a tree does not affect the Neumann eigenvalues. However, increasing an interior vertex to a tree reduces the value of corresponding Neumann eigenvalues. As a consequence of this result, we provide an upper bound for the second Neumann eigenvalue and a lower bound for the largest Neumann eigenvalue on trees. Then, we obtain a sharp upper bound for the second Neumann eigenvalue on paths in terms of its diameter, and as an application, we show that the second Neumann eigenvalue cannot be bounded below by a positive real number on the family of paths. We also prove that under a diameter constraint on trees, the largest Neumann eigenvalue cannot be bounded from above. Finally, we obtain a lower bound for the second Neumann eigenvalue on graphs.