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Computing the $p$-Laplacian eigenpairs of signed graphs

Chuanyuan Ge, Ouyuan Qin

TL;DR

A new criterion is provided to determine when a graph is not a subgraph of another one, which outperforms existing criteria based on the linear Laplacian and adjacency matrices and highlights the deep connections and numerous similarities between the spectral theories of tensors and graph $p-Laplacians.

Abstract

As a nonlinear extension of the graph Laplacian, the graph $p$-Laplacian has various applications in many fields. Due to the nonlinearity, it is very difficult to compute the eigenvalues and eigenfunctions of graph $p$-Laplacian. In this paper, we establish the equivalence between the graph $p$-Laplacian eigenproblem and the tensor eigenproblem when $p$ is even. Building on this result, algorithms designed for tensor eigenproblems can be adapted to compute the eigenpairs of the graph $p$-Laplacian. For general $p>1$, we give a fast and convergent algorithm to compute the largest eigenvalue and the corresponding eigenfunction of the signless graph $p$-Laplacian. As an application, we provide a new criterion to determine when a graph is not a subgraph of another one, which outperforms existing criteria based on the linear Laplacian and adjacency matrices. Our work highlights the deep connections and numerous similarities between the spectral theories of tensors and graph $p$-Laplacians.

Computing the $p$-Laplacian eigenpairs of signed graphs

TL;DR

A new criterion is provided to determine when a graph is not a subgraph of another one, which outperforms existing criteria based on the linear Laplacian and adjacency matrices and highlights the deep connections and numerous similarities between the spectral theories of tensors and graph $p-Laplacians.

Abstract

As a nonlinear extension of the graph Laplacian, the graph -Laplacian has various applications in many fields. Due to the nonlinearity, it is very difficult to compute the eigenvalues and eigenfunctions of graph -Laplacian. In this paper, we establish the equivalence between the graph -Laplacian eigenproblem and the tensor eigenproblem when is even. Building on this result, algorithms designed for tensor eigenproblems can be adapted to compute the eigenpairs of the graph -Laplacian. For general , we give a fast and convergent algorithm to compute the largest eigenvalue and the corresponding eigenfunction of the signless graph -Laplacian. As an application, we provide a new criterion to determine when a graph is not a subgraph of another one, which outperforms existing criteria based on the linear Laplacian and adjacency matrices. Our work highlights the deep connections and numerous similarities between the spectral theories of tensors and graph -Laplacians.
Paper Structure (6 sections, 11 theorems, 32 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 6 sections, 11 theorems, 32 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivalence of signed graph p-Laplacian eigenproblem and tensor eigenproblem for even p
  4. An algorithm based on Perron--Frobenius type theorem
  5. Application
  6. Further remarks

Key Result

Proposition 3.1

\newlabelprop:tensor0 Let $\Gamma=(G,\sigma)$ be a signed graph with $G=(V,E,w,\mu,\kappa)$. Assume $p \geq 2$ is an even number and $i$ and $j$ are arbitrary vertices in $V$. Define two symmetric $p$-th order $n$-dimension tensors $\mathcal{T}^{(p)}$ and $\mathcal{B}^{(p)}$ as follows and where $|\{i_1,i_2,\ldots,i_{p}\}|$ is the cardinality of set $\{i_1,i_2,\ldots,i_{p}\}$ and $\{i_1,i_2,\ld

Figures (2)

  • Figure 1: Left: relative gap between $\overline{\lambda}_{k}$ and $\underline{\lambda}_{k}$ versus iteration for random initial vectors. Right: time for solving the largest eigenvalue of signless 20-Laplacian of different graphs.
  • Figure 1: Left: the graph $G'$ used in Example 5.1. Right: the largest eigenvalues of signless $p$-Laplacian of $G$ and $G'$ for various $p$.

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4: CDN
  • Remark 2.5
  • Proposition 3.1
  • Proof 1
  • Lemma 3.2
  • Theorem 3.3
  • Example 3.4
  • ...and 23 more