Maximization of the first Laplace eigenvalue of a finite graph II

TL;DR

This paper investigates the maximization of the first nonzero eigenvalue of the Fujiwara Laplacian on finite graphs, proving that the supremum is infinite for graphs containing cycles when edge lengths are optimized.

Contribution

It establishes that the maximum first eigenvalue is unbounded for cyclic graphs under edge-length normalization, extending previous work on spectral optimization.

Findings

Supremum of the first eigenvalue is infinite for graphs with cycles.

Edge-length optimization can lead to arbitrarily large eigenvalues.

Results apply to Fujiwara Laplacian on finite graphs.

Abstract

Given a length function on the set of edges of a finite graph, the corresponding Fujiwara Laplacian is defined. We consider a problem of maximizing the first nonzero eigenvalue of this graph Laplacian over all choices of edge-length function subject to a certain normalization. In this paper we prove that the supremum of the first nonzero eigenvalue is infinite whenever the graph contains a cycle.