Proof of a conjecture on ‘plateaux’ phenomenon of graph Laplacian eigenvalues

Tree simplification and the 'plateaux' phenomenon of graph Laplacian eigenvalues

We developed a procedure of reducing the number of vertices and edges of a given tree, which we call the " tree simplification procedure, " without changing its topological information. Our motivation for developing this procedure was to reduce computational costs of graph Laplacian eigenvalues of such trees. When we applied this procedure to a set of trees representing dendritic structures of ...

A lower bound for the Laplacian eigenvalues of a graph—proof of a conjecture by Guo

We show that if μj is the j-th largest Laplacian eigenvalue, and dj is the j-th largest degree (1 ≤ j ≤ n) of a connected graph Γ on n vertices, then μj ≥ dj − j + 2 (1 ≤ j ≤ n− 1). This settles a conjecture due to Guo.

On Laplacian Eigenvalues of a Graph

Let G be a connected graph with n vertices and m edges. The Laplacian eigenvalues are denoted by μ1(G) ≥ μ2(G) ≥ ·· · ≥ μn−1(G) > μn(G) = 0. The Laplacian eigenvalues have important applications in theoretical chemistry. We present upper bounds for μ1(G)+ · · ·+μk(G) and lower bounds for μn−1(G)+ · · ·+μn−k(G) in terms of n and m, where 1 ≤ k ≤ n−2, and characterize the extremal cases. We also ...

Laplacian eigenvalues and optimality: II. The Laplacian of a graph

On the Distribution of Laplacian Eigenvalues of a Graph

This paper presents some bounds on the number of Laplacian eigenvalues contained in various subintervals of [0, n] by using the matching number and edge covering number for G, and asserts that for a connected graph the Laplacian eigenvalue 1 appears with certain multiplicity. Furthermore, as an application of our result (Theorem 13), Grone and Merris’ conjecture [The Laplacian spectrum of graph...