Petals and Books: The largest Laplacian spectral gap from 1

TL;DR

This paper establishes the maximum spectral gap from 1 for the normalized Laplacian in connected graphs, identifying specific extremal graph structures and implications for eigenvalue convergence rates.

Contribution

It proves the maximum spectral gap from 1 is 1/2 for connected graphs and characterizes extremal graphs as petal or book graphs, linking to known bounds.

Findings

Maximum spectral gap from 1 is 1/2 for connected graphs.

Extremal graphs are petal graphs (odd N) and book graphs (even N).

Provides bounds for eigenvalue convergence rates.

Abstract

We prove that, for any connected graph on $N\geq 3$ vertices, the spectral gap from the value $1$ with respect to the normalized Laplacian is at most $1/2$. Moreover, we show that equality is achieved if and only if the graph is either a petal graph (for $N$ odd) or a book graph (for $N$ even). This implies that $(\frac12,\frac32)$ is a maximal gap interval for the normalized Laplacian on connected graphs. This is closely related to the Alon-Boppana bound on regular graphs and a recent result by Koll\'ar and Sarnak on cubic graphs. Our result also provides a sharp bound for the convergence rate of some eigenvalues of the Laplacian on neighborhood graphs.