Spectral gap of the normalized distance Laplacian

Abstract

The smallest positive eigenvalue $\partial_2$ of the normalized distance Laplacian matrix $\mathcal{D}^{\mathcal{L}}$ of a connected graph is called its \emph{spectral gap} and is intimately related to the Cheeger constant of $\mathcal{D}^{\mathcal{L}}$. Byrne, Johnston, Schildkraut and Tait (2025) conjectured that \[ \partial_2 \ge \frac{2}{3}\] for all connected graphs. We prove the following stronger result: for any connected graph $G$ of order at least 2, \[\partial_2 \ge \frac{2}{3} + \frac{4}{3\,t_{\max}},\] where $t_{\max}$ denotes the maximum transmission in $G$. Moreover, equality holds if and only if $G\cong K_{m,m}$ for some $m\ge 1$.

Disclosure

“= Km,m + − where m = | supp (y)| = | supp (y)|. Combining Claims 2.2 and 2.3 proves Theorem 1.3. AI statement We acknowledge the use of AI tools during the ideation phase. We declare that the text is not AI-generated. References [1] Sinan G. Aksoy, Fan Chung, Michael Tait, and Josh Tobin. The maximum relaxation time of a random walk. Adv. in Appl. Math., 101:1–14, 2018. 3”

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Classification

Brainstorming or outlining

Multiplier

2

Verified

2026-08-23T05:09:17.729502Z

Structural counts