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# Mathematics > Spectral Theory

# Title: Homological eigenvalues of graph $p$-Laplacians

(Submitted on 12 Oct 2021 (v1), last revised 22 Nov 2021 (this version, v3))

Abstract: We introduce the homological eigenvalues of the graph $p$-Laplacian $\Delta_p$, and we prove that for any homological eigenvalue $\lambda(\Delta_p)$, the functions $p(2\lambda(\Delta_p))^{\frac1p}$ and $2^{-p}\lambda(\Delta_p)$ are locally increasing and decreasing with respect to $p$, respectively. We show the existence of non-homological eigenvalue of $\Delta_p$, and for general eigenvalues, the local monotonicity doesn't hold, but we have the upper semi-continuity of the spectra of graph $p$-Laplacians with respect to $p$. The $k$-th min-max eigenvalue $\lambda_k(\Delta_p)$ is a homological eigenvalue for any $k=1,2,\cdots$, and we establish the locally Lipschitz continuity and monotonicity of $p(2\lambda_k(\Delta_p))^{\frac1p}$ and $2^{-p}\lambda_k(\Delta_p)$ with respect to $p\in[1,+\infty)$. Moreover, we provide a homological eigenvalue which is not of min-max form.

These discoveries establish asymptotic behaviors of $\Delta_p$-eigenvalues for varying $p$, which lead to several applications, including: (1) settle an open problem on the monotonicity of a certain function related to the eignevalues of graph $p$-Laplacians with respect to $p$; (2) resolve a question asking whether the third eigenvalue of graph $p$-Laplacian is of min-max form; (3) refine the higher order Cheeger inequalities for graph $p$-Laplacians.

## Submission history

From: Dong Zhang [view email]**[v1]**Tue, 12 Oct 2021 14:57:04 GMT (24kb)

**[v2]**Mon, 18 Oct 2021 15:55:12 GMT (25kb)

**[v3]**Mon, 22 Nov 2021 21:18:20 GMT (34kb)

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