Related papers: Beyond symmetrization: effective adjacency matrice…
It has been an open problem to find the Moore-Penrose inverses of the incidence, Laplacian, and signless Laplacian matrices of families of graphs except trees and unicyclic graphs. Since the inverse formulas for an odd unicyclic graph and…
We propose an algebraic framework for generalized graph Laplacians which unifies the study of resistor networks, the critical group, and the eigenvalues of the Laplacian and adjacency matrices. Given a graph with boundary $G$ together with…
The aim of the present paper is to analyse the spectrum of Laplace and Dirac type operators on metric graphs. In particular, we show for equilateral graphs how the spectrum (up to exceptional eigenvalues) can be described by a natural…
The representation of graphs is commonly based on the adjacency matrix concept. This formulation is the foundation of most algebraic and computational approaches to graph processing. The advent of deep learning language models offers a wide…
In this paper, we introduce the Laplacian and the signless Laplacian for the eccentricity matrix of a connected graph, referred to as the eccentricity Laplacian and the eccentricity signless Laplacian, respectively. We establish the…
In this study, we challenge the traditional approach of frequency analysis on directed graphs, which typically relies on a single measure of signal variation such as total variation. We argue that the inherent directionality in directed…
The pseudo-inverse of a graph Laplacian matrix, denoted as $L^\dagger$, finds extensive application in various graph analysis tasks. Notable examples include the calculation of electrical closeness centrality, determination of Kemeny's…
Resistance distance has been studied extensively in the past years, with the majority of previous studies devoted to undirected networks, in spite of the fact that various realistic networks are directed. Although several generalizations of…
For a compact, connected metric graphs with a boundary that consists of $k$ vertices, we prove that an arbitrary symmetric $k\times k$ matrix with real entries can be realized as the Dirichlet-to-Neumann operator for the Laplacian plus a…
A common approach for analyzing hypergraphs is to consider the projected adjacency or Laplacian matrices for each order of interactions (e.g., dyadic, triadic, etc.). However, this method can lose information about the hypergraph structure…
Though commonly found in the real world, directed networks have received relatively less attention from the literature in which concerns their topological and dynamical characteristics. In this work, we develop a magnetic Laplacian-based…
The Helmholtzian matrix of a graph $G=(V(G),E(G))$ is a graph-theoretic analogue of the vector Laplacian (or Helmholtz operator) [S. Li, L. Lu, J.F. Wang, A graph discretization of vector Laplacian, 379 (2026) 446--460]. Motivated by the…
Operators are induced on fermion and zeon algebras by the action of adjacency matrices and combinatorial Laplacians on the vector spaces spanned by the graph's vertices. Properties of the algebras automatically give information about the…
Image deblurring is relevant in many fields of science and engineering. To solve this problem, many different approaches have been proposed and among the various methods, variational ones are extremely popular. These approaches are…
The Laplacian matrix of a simple graph is the difference of the diagonal matrix of vertex degree and the (0,1) adjacency matrix. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been…
Recent research has tried to extend the concept of renormalization, which is naturally defined for geometric objects, to more general networks with arbitrary topology. The current attempts do not naturally apply to directed networks, for…
The pseudoinverse of a graph Laplacian is used in many applications and fields, such as for instance in the computation of the effective resistance in electrical networks, in the calculation of the hitting/commuting times for a Markov chain…
Because of the significant increase in size and complexity of the networks, the distributed computation of eigenvalues and eigenvectors of graph matrices has become very challenging and yet it remains as important as before. In this paper…
Chemical hypergraphs and their associated normalized Laplace operators are generalized and studied in the case where each vertex--hyperedge incidence has a real coefficient. We systematically study the effect of symmetries of a hypergraph…
We consider a magnetic Laplacian with periodic magnetic potentials on periodic discrete graphs. Its spectrum consists of a finite number of bands, where degenerate bands are eigenvalues of infinite multiplicity. We obtain a specific…