Nonlinear spectral graph theory

TL;DR

We study nonlinear spectral graph theory centered on the graph $p$-Laplacian, defining nonlinear eigenpairs via critical points of the Rayleigh quotient $\mathcal{R}_p(f)=\|Kf\|_p/\|f\|_p$ and investigating how spectrum interacts with graph topology. The paper surveys multiple variational frameworks (Krasnoselskii, Drábek, Yang, and $C^2$-sphere) and extends to degenerate cases ($p=1,\infty$) with viscosity, Clarke, and generalized eigenpairs, while establishing duality between node and edge formulations. New results connect the $\infty$-Laplacian spectrum to sphere packing radii, independence and matching numbers, and perfect nodal domains, and relate to minimal $k$-partitions, Dirichlet Cheeger constants, and higher-order Cheeger constants. Regularity results show continuity in $p$, convergence to $\Delta_\infty$ spectra, and a nuanced nodal-domain theory across $p$, with broad implications for graph clustering, partitioning, and core-periphery analysis in nonlinear spectral settings.

Abstract

Nonlinear spectral graph theory is an extension of the traditional (linear) spectral graph theory and studies relationships between spectral properties of nonlinear operators defined on a graph and topological properties of the graph itself. Many of these relationships get tighter when going from the linear to the nonlinear case. In this manuscript, we discuss the spectral theory of the graph $p$-Laplacian operator. In particular we report links between the $p$-Laplacian spectrum and higher-order Cheeger (or isoperimetric) constants, sphere packing constants, independence and matching numbers of the graph. The main aim of this paper is to present a complete and self-contained introduction to the problem accompanied by a discussion of the main results and the proof of new results that fill some gaps in the theory. The majority of the new results are devoted to the study of the graph infinity Laplacian spectrum and the information that it yields about the packing radii, the independence numbers and the matching number of the graph. This is accompanied by a novel discussion about the nodal domains induced by the infinity eigenfunctions. There are also new results about the variational spectrum of the $p$-Laplacian, the regularity of the $p$-Laplacian spectrum varying $p$, and the relations between the $1$-Laplacian spectrum and new Cheeger constants.

Figures (7)

• Figure 1: We consider a complete graph on three vertices, with edge weights $1^{1/p}$, $1.5^{1/p}$, and $5^{1/p}$. The corresponding Rayleigh quotient: $\mathcal{R}_p^p(x) = (5|x_1 - x_2|^p + |x_1 - x_3|^p + 1.5|x_2 - x_3|^p)/(|x_1|^p + |x_2|^p + |x_3|^p)$ is plotted from left to right for $p = 2$ in the projective space $x_2 \neq 0$, for $p = 6$ in the projective space $x_2 \neq 0$, and for $p = 6$ in the projective space $x_3 \neq 0$. In the linear case ($p = 2$), we observe the existence of only three critical points: one maximum, one minimum, and one saddle point. Conversely, in the nonlinear case ($p = 6$), we observe three saddle points, three maxima, and one minimum at $x_1 = x_2 = x_3$, yielding a total of seven distinct eigenvalues.
• Figure 2: Left: Example graph in which the corresponding $p$-Laplacian $\Delta_p$ with $\omega_{uv}=1 \;\forall (u,v)\in E$, has more eigenvalues than the dimension of the space. Right: Set of five eigenvalues and corresponding eigenfunctions.
• Figure 3: From left to right and from top to bottom, we report the eigenspaces relative to the $p$-Laplacian eigenvalues for $p$ taking values $1,2,3$ and $\infty$. Dashed lines denote the vectors $f_{i,j}$ varying $i$ and $j$ in $\{1,2,3\}$. Similarly, for $p=3$, the solid red lines denote the vectors $g_i$ and $-g_i$ for $i=1,2,3$. Finally the red regions denote the eigenspaces relative to $\Lambda_{a,b}(\Delta_p)$, different opacities correspond to different supporting planes.
• Figure 4:
• Figure 5:

Theorems & Definitions (148)

• Definition 2.1
• Definition 3.1
• Remark 3.2
• Definition 3.3: Unweighted graph
• Lemma 3.5: Deformation Lemma
• Theorem 3.6
• Definition 3.7
• Definition 3.8
• Lemma 3.9
• Proposition 3.10