• Corpus ID: 244488736

Homological eigenvalues of graph $p$-Laplacians

  title={Homological eigenvalues of graph \$p\$-Laplacians},
  author={Dong Zhang},
We introduce the homological eigenvalues of the graph p-Laplacian ∆p, and we prove that for any homological eigenvalue λ(∆p), the functions p(2λ(∆p)) 1 p and 2λ(∆p) are locally increasing and decreasing with respect to p, respectively. We show the existence of non-homological eigenvalue of ∆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 λk(∆p) is a… 

Figures from this paper

Nodal domain count for the generalized graph $p$-Laplacian
It is shown how to transfer Weyl’s inequalities for the Laplacian operator to the nonlinear case and prove new upper and lower bounds on the number of nodal domains of every eigenfunction of the generalized p-LaPLacian on generic graphs, including variational eigenpairs.


Spectrum of the 1-Laplacian and Cheeger's Constant on Graphs
  • K. Chang
  • Computer Science, Mathematics
    J. Graph Theory
  • 2016
A nonlinear spectral graph theory is developed, in which the Laplace operator is replaced by the 1 - Laplacian Δ1, and Cheeger's constant equals to the first nonzero Δ1 eigenvalue for connected graphs.
A nodal domain theorem and a higher-order Cheeger inequality for the graph $p$-Laplacian
A nodal domain theorem is proved for the graph p-Laplacian for any $p\geq 1$ and the higher order Cheeger inequality becomes tight as $ p\rightarrow 1$.
Nodal domains of eigenvectors for 1-Laplacian on graphs
Abstract The eigenvectors for graph 1-Laplacian possess some sort of localization property: On one hand, the characteristic function on any nodal domain of an eigenvector is again an eigenvector with
Multi-way spectral partitioning and higher-order cheeger inequalities
This work shows that in every graph there are at least k/2 disjoint sets, each having expansion at most O(√(λk log k), and proves that the √(log k) bound is tight, up to constant factors, for the "noisy hypercube" graphs.
Topological multiplicity of the maximum eigenvalue of graph 1-Laplacian
  • Dong Zhang
  • Mathematics, Computer Science
    Discret. Math.
  • 2018
The pseudo independent number of a graph is introduced, which provides a better lower estimate of the topological multiplicity of the maximum eigenvalue of the corresponding graph 1-Laplacian.
Optimization results for the higher eigenvalues of the p‐Laplacian associated with sign‐changing capacitary measures
In this paper we prove the existence of an optimal set for the minimization of the $k$-th variational eigenvalue of the $p$-Laplacian among $p$-quasi open sets of fixed measure included in a box of
Spectral clustering based on the graph p-Laplacian
The experiments show that the clustering found by p-spectral clustering is at least as good as normal spectral clustering, but often leads to significantly better results.
On eigenfunctions of Markov processes on trees
We begin by studying the eigenvectors associated to irreducible finite birth and death processes, showing that the i nontrivial eigenvector φi admits a succession of i decreasing or increasing
Homological illusions of persistence and stability
In this thesis we explore and extend the theory of persistent homology, which captures topological features of a function by pairing its critical values. The result is represented by a collection of
Categorification of Persistent Homology
This work redevelops persistent homology (topological persistence) from a categorical point of view and gives a natural construction of a category of ε-interleavings of $\mathbf {(\mathbb {R},\leq)}$-indexed diagrams in some target category and shows that if the target category is abelian, so is this category of interleavments.