Maximization of the first Laplace eigenvalue of a finite graph

TL;DR

This paper investigates how to maximize the first nonzero eigenvalue of a graph Laplacian by optimizing edge lengths, providing explicit formulas and analogies to smooth surface eigenvalue problems.

Contribution

It establishes a characterization of extremal solutions for eigenvalue maximization on graphs, extending results from smooth surfaces to discrete graphs.

Findings

Existence of a vertex map related to first eigenfunctions

Explicit expression of edge lengths in terms of eigenfunctions

Extension of results to G"oring-Helmberg-Wappler problem

Abstract

Given a length function on the edge set of a finite graph, we define a vertex-weight and an edge-weight in terms of it and consider the corresponding graph Laplacian. In this paper, we consider the problem of maximizing the first nonzero eigenvalue of this Laplacian over all edge-length functions subject to a certain normalization. For an extremal solution of this problem, we prove that there exists a map from the vertex set to a Euclidean space consisting of first eigenfunctions of the corresponding Laplacian so that the length function can be explicitly expressed in terms of the map and the Euclidean distance. This is a graph-analogue of Nadirashvili's result related to first-eigenvalue maximization problem on a smooth surface. We discuss simple examples and also prove a similar result for a maximizing solution of the G\"oring-Helmberg-Wappler problem.