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arXiv · 2110.08751

Petals and Books: The largest Laplacian spectral gap from 1

Jürgen Jost·Raffaella Mulas·Dong Zhang
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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.

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Jürgen Jost, Raffaella Mulas, Dong Zhang. 2021-10-17. Petals and Books: The largest Laplacian spectral gap from 1. https://doi.org/10.1002/jgt.22999

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