Related papers: Analysis of Dirichlet forms on graphs
This is a survey paper. We study the Ricci curvature and spectrum of graphs, as well as the exterior forms and deRahm cohomology on graphs.
The paper investigates relationship between algebraic expressions and graphs. We consider a digraph called a Fibonacci graph which gives a generic example of non-series-parallel graphs. Our intention in this paper is to simplify the…
In this paper we survey many results on the Dirichlet space of analytic functions. Our focus is more on the classical Dirichlet space on the disc and not the potential generalizations to other domains or several variables. Additionally, we…
We study the relation between algebraic structures and Graph Theory. We have defined five different weighted digraphs associated to a finite dimensional algebra over a field in order to tackle important properties of the associated…
The fundamental role of on-shell diagrams in quantum field theory has been recently recognized. On-shell diagrams, or equivalently bipartite graphs, provide a natural bridge connecting gauge theory to powerful mathematical structures such…
This paper is devoted to the study of directed graphs with extremal properties relative to certain metric functionals. We characterize up to isomorphism critical digraphs with infinite values of diameter, quasi-diameter, radius and…
A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint…
Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information.…
We survey the relationship between the combinatorics and geometry of graphs and the algebraic structure of right-angled Artin groups. We concentrate on the defining graph of the right-angled Artin group and on the extension graph associated…
This is a survey article written for the Jahresberichte der DMV. Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and…
We study spaces of realisations of linkages (weighted graphs) whose underlying graph is a series parallel graph. In particular, we describe an algorithm for determining whether or not such spaces are connected.
We investigate the relationship between measurable differentiable structures on doubling metric measure spaces and derivations. We prove: [1] a decomposition theorem for the module of derivations into free modules; [2] the existence of a…
We introduce and investigate generalizations of interval and proper interval graphs to simplicial complexes, including strong interval, unit interval, and under closed variants. Through equivalent combinatorial and algebraic…
This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active…
In this work we study the intrinsic geometry of the space of Kahler metrics under various Riemannian metrics. The first part is on the Dirichlet metric. We motivate its study, we compute its curvature, and we make links with the Calabi…
This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose a curvature on the nodes and links of a graph and present the evidence for their interpretation as a curvature. Notably,…
There is a well-documented research programme on graph operators which addresses questions such as `Which graphs appear as images of graphs?'; `Which graphs are fixed under the operator?'; `What happens if the operator is iterated?' In this…
In this article we are introducing combinatorial spectra of graphs, this is a generalization of $H$-Hamiltonian spectra. The main motivation was to made from $H$-Hamiltonian spectra an operation and develop some algebra in this field. An…
This paper studies \emph{Dirichlet arrangements}, a generalization of graphic hyperplane arrangements arising from electrical networks and order polytopes of finite posets. We generalize descriptions of combinatorial features of graphic…
In this paper we study the analytic properties of a multiple Dirichlet series associated to the prehomogeneous vector space of binary cubic forms.