Sharp bounds and monotonicity results for Neumann eigenvalues

TL;DR

This paper establishes sharp bounds and monotonicity properties for Neumann eigenvalues of the Laplace operator on graphs, with specific results for trees and paths, advancing understanding of spectral graph theory.

Contribution

It introduces new monotonicity results for Neumann eigenvalues on trees and provides sharp bounds for these eigenvalues under various graph constraints.

Findings

Increasing boundary vertices does not affect Neumann eigenvalues.

Adding interior vertices reduces Neumann eigenvalues.

Sharp bounds for the second and largest Neumann eigenvalues on trees and paths.

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.