Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Homological eigenvalues of graph $p$-Laplacians   Inspired by persistent homology in topological data analysis, we introduce
the homological eigenvalues of the graph $p$-Laplacian $\Delta_p$, which allows
us to analyse and classify non-variational eigenvalues. We show the stability
of homological eigenvalues, and we prove that for any homological eigenvalue
$\lambda(\Delta_p)$, the function $p\mapsto p(2\lambda(\Delta_p))^{\frac1p}$ is
locally increasing, while the function $p\mapsto 2^{-p}\lambda(\Delta_p)$ is
locally decreasing. As a special class of homological eigenvalues, the
variational (min-max) eigenvalues $\lambda_1(\Delta_p)$, $\lambda_2(\Delta_p)$,
$\cdots$, $\lambda_k(\Delta_p)$, $\cdots$, are locally Lipschitz continuous
with respect to $p\in[1,+\infty)$. We also establish the monotonicity of
$p(2\lambda_k(\Delta_p))^{\frac1p}$ and $2^{-p}\lambda_k(\Delta_p)$ with
respect to $p\in[1,+\infty)$.
  These results establish refined analysis of $\Delta_p$-eigenvalues for
varying $p$, which lead to several applications, including: (1) settle an open
problem by Amghibech on the monotonicity of some function involving eigenvalues
of $p$-Laplacian 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 by Tudisco and Hein,
and extend the multi-way Cheeger inequality by Lee, Oveis Gharan and Trevisan
to the $p$-Laplacian case.
  Furthermore, for the 1-Laplacian case, we characterize the homological
eigenvalues and variational eigenvalues from the perspective of topological
combinatorics, where our idea is similar to the authors' work on discrete Morse
theory.

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The categories of this paper are: nucl-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: physics
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The categories of this paper are: cs
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The categories of this paper are: stat
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-th
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The categories of this paper are: mtrl-th