A Universal upper bound on Graph Diameter based on Laplacian Eigenvalues

TL;DR

This paper establishes a universal upper bound on the diameter of any unweighted connected graph using Laplacian eigenvalues, specifically relating it to the k-th smallest eigenvalue, addressing a problem posed by Gil Kalai.

Contribution

The authors prove a new bound on graph diameter based on Laplacian eigenvalues, providing a solution to an open problem by Gil Kalai.

Findings

Diameter is O(k log n / lambda_k) for any unweighted connected graph.

The bound links spectral properties to graph diameter.

Addresses a previously open problem in spectral graph theory.

Abstract

We prove that the diameter of any unweighted connected graph G is O(k log n/lambda_k), for any k>= 2. Here, lambda_k is the k smallest eigenvalue of the normalized laplacian of G. This solves a problem posed by Gil Kalai.