Related papers: Planarity, duality and Laplacian congruence

A speculative overview of a future topic of research. The paper is a collection of ideas concerning two related areas: 1) Graph computation machines ("computing with graphs"). This is the class of models of computation in which the state of…

In this note we elaborate on some notions of surface area for discrete graphs which are closely related to the inverse degree. These notions then naturally lead to associated connectivity measures of graphs and to the definition of a…

Coalescing involves gluing one or more rooted graphs onto another graph. Under specific conditions, it is possible to start with cospectral graphs that are coalesced in similar ways that will result in new cospectral graphs. We present a…

We consider the matrix regularization of scalar fields on a Riemann surface with a general gauge-field background. We propose a construction of the fuzzy version of the Laplacian.

By using Alexander duality on simplicial complexes we give a new and algebraic proof of Dirac's theorem on chordal graphs.

In [Steve Butler. A note about cospectral graphs for the adjacency and normalized Laplacian matrices. Linear Multilinear Algebra, 58(3-4):387-390, 2010.], Butler constructed a family of bipartite graphs, which are cospectral for both the…

We study operators on rooted graphs with a certain spherical homogeneity. These graphs are called path commuting and allow for a decomposition of the adjacency matrix and the Laplacian into a direct sum of Jacobi matrices which reflect the…

In this paper, we give the spectrum of a matrix by using the quotient matrix, then we apply this result to various matrices associated to a graph and a digraph, including adjacency matrix, (signless) Laplacian matrix, distance matrix,…

This chapter is an introduction to the connection between random matrices and maps, i.e graphs drawn on surfaces. We concentrate on the one-matrix model and explain how it encodes and allows to solve a map enumeration problem.

In spectral graph theory, Grone and Merris conjecture that the spectrum of the Laplacian matrix of a finite graph is majorized by the conjugate degree sequence of this graph. We give a complete proof for this conjecture.

The notion of 1-planarity is among the most natural and most studied generalizations of graph planarity. A graph is 1-planar if it has an embedding where each edge is crossed by at most another edge. The study of 1-planar graphs dates back…

We establish connectedness criteria for graphs associated to monomials in certain quotients of the mod 2 dual Steenrod algebra. We also investigate questions about trees and Hamilton cycles in the context of these graphs. Finally, we…

In this paper we consider graphs whose edges are associated with a degree of {\em importance}, which may depend on the type of connections they represent or on how recently they appeared in the scene, in a streaming setting. The goal is to…

In an earlier work, the author together with Guo [Hermitian adjacency matrix of digraphs and mixed graphs, J. Graph Theory 85 (2017) 217-248] introduced the Hermitian adjacency matrix of directed (and partially directed) graphs. However, it…

We study the surface of Gauss double points associated to a very general quartic surface and the natural morphisms associated to it.

We study topological properties of the graph topology.

Aiming at a generalization of a classical theorem of Moebius, we study maps that take line intervals to plane curves, and also maps that take line intervals to conics from certain linear systems.

This paper further investigates \emph{Laplacian simplices}. A construction by Braun and the first author associates to a simple connected graph $G$ a simplex $\cP_G$ whose vertices are the rows of the Laplacian matrix of $G$. In this paper…

The properties of a hypergraph explored through the spectrum of its unified matrix was made by the authors in [26]. In this paper, we introduce three different hypergraph matrices: unified Laplacian matrix, unified signless Laplacian…

For a given graph $\mathcal{G}$ of order $n$ with $m$ edges, and a real symmetric matrix associated to the graph, $M\left(\mathcal{G}\right)\in\mathbb{R}^{n\times n}$, the interlacing graph reduction problem is to find a graph…