The graph $\infty$-Laplacian eigenvalue problem
SIAM Journal on Mathematical Analysis, 2026
We analyze various formulations of the $\infty$-Laplacian eigenvalue problem on graphs, comparing their properties and highlighting their respective advantages and limitations. First, we investigate the graph $\infty$-eigenpairs arising as limits of $p$-Laplacian eigenpairs, extending key results from the continuous …
Nonlinear Spectral Duality
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2026
Nonlinear eigenvalue problems for pairs of homogeneous convex functions are particular nonlinear constrained optimization problems that arise in a variety of settings, including graph mining, machine learning, and network science. By considering different notions of duality transforms from both classical and recent …
Nodal domain count for the generalized graph $p$-Laplacian
Applied and Computational Harmonic Analysis, vol. 64, pp. 1–32, 2023
Inspired by the linear Schrödinger operator, we consider a generalized $p$-Laplacian operator on discrete graphs and present new results that characterize several spectral properties of this operator with particular attention to the nodal domain count of its eigenfunctions. Just like the one-dimensional continuous …
The self-consistent field iteration for p-spectral clustering
The self-consistent field (SCF) iteration, combined with its variants, is one of the most widely used algorithms in quantum chemistry. We propose a procedure to adapt the SCF iteration for the p-Laplacian eigenproblem, which is an important problem in the field of unsupervised learning. We formulate the p-Laplacian …
Modularity bounds for clusters located by leading eigenvectors of the normalized modularity matrix
Journal of Mathematical Inequalities, vol. 11, pp. 701–714, 2017
Nodal theorems for generalized modularity matrices ensure that the cluster located by the positive entries of the leading eigenvector of various modularity matrices induces a connected subgraph. In this paper we obtain lower bounds for the modularity of that set of nodes showing that, under certain conditions, the …
A nodal domain theorem and a higher-order Cheeger inequality for the graph p-Laplacian
EMS Journal of Spectral Theory, vol. 8, pp. 883–908, 2018
We consider the nonlinear graph $p$-Laplacian and its set of eigenvalues and associated eigenfunctions of this operator defined by a variational principle. We prove a nodal domain theorem for the graph $p$-Laplacian for any $p\geq 1$. While for $p>1$ the bounds on the number of weak and strong nodal domains are the …
An algebraic analysis of the graph modularity
SIAM Journal on Matrix Analysis and Applications, vol. 35, pp. 997–1018, 2014
One of the most relevant tasks in network analysis is the detection of community structures, or clustering. Most popular techniques for community detection are based on the maximization of a quality function called modularity, which in turn is based upon particular quadratic forms associated to a real symmetric …