Related papers: Analysis of Dirichlet forms on graphs
Graph-based signal processing techniques have become essential for handling data in non-Euclidean spaces. However, there is a growing awareness that these graph models might need to be expanded into `higher-order' domains to effectively…
There are versions of "calculus" in many settings, with various mixtures of algebra and analysis. In these informal notes we consider a few examples that suggest a lot of interesting questions.
The class of closed graphs by a linear ordering on their sets of vertices is investigated. A recent characterization of such a class of graphs is analyzed by using tools from the proper interval graph theory.
A new kind of diagrams is presented, showing the causal structure of bimetric interactions.
We adapt the study of hyperspaces and function spaces from classical topology to digital topology. We define digital hyperspaces and digital function graphs, and study some of their relationships and graphical properties.
In this paper we study a construction of algebraic curves from combinatorial data. In the study of algebraic curves through degeneration, graphs usually appear as the dual intersection graph of the central fiber. Properties of such graphs…
We suggest a combinatorial classification of metric filtrations in measure spaces; a complete invariant of such a filtration is its combinatorial scheme, a measure on the space of hierarchies of the group~$\mathbb Z$. In turn, the notion of…
We develop a theory of bounded variation functions and Besov spaces in abstract Dirichlet spaces which unifies several known examples and applies to new situations, including fractals.
The topic of this treatise is a combinatorial technique called Graph Pebbling. We investigate pebbling numbers, weight functions, flow networks, hypercubes, and the zero-sum conjecture of Erd\H{o}s and Lemke. This investigation is a…
This thesis deals with the enumerative study of combinatorial maps, and its application to the enumeration of other combinatorial objects. Combinatorial maps, or simply maps, form a rich combinatorial model. They have an intuitive and…
Metric graphs are meaningful objects for modeling complex structures that arise in many real-world applications, such as road networks, river systems, earthquake faults, blood vessels, and filamentary structures in galaxies. To study metric…
We define and study structural properties of hypergraphs of models of a theory including lattice ones. Characterizations for the lattice properties of hypergraphs of models of a theory, as well as for structures on sets of isomorphism types…
In this paper, we extend the recently introduced concept of partially dual ribbon graphs to graphs. We then go on to characterize partial duality of graphs in terms of bijections between edge sets of corresponding graphs. This result…
The concept of coreflexive set is introduced to study the structure of digraphs. New characterizations of line digraphs and nth-order line digraphs are given. Coreflexive sets also lead to another natural way of forming an intersection…
Using the technique of the metrization theorem of uniformities with countable bases, in this note we provide, test and compare an explicit algorithm to produce a metric $d(x,y)$ between the vertices $x$ and $y$ of an affinity weighted…
The study of combinatorial games is intimately tied to the study of graphs, as any game can be realized as a directed graph in which players take turns traversing the edges until reaching a sink. However, there have heretofore been few…
In this (mostly) survey article, we give a synopsis of a number of results relating to Brill--Noether theory on curves and metric graphs, together with some speculations about the behavior of one-dimensional linear series on a class of…
There are several interrelated notions of discrete curvature on graphs. Many approaches utilize the optimal transportation metric on its probability simplex or the distance matrix of the graph. In this survey article, we compute formulas…
Existing approaches to solving combinatorial optimization problems on graphs suffer from the need to engineer each problem algorithmically, with practical problems recurring in many instances. The practical side of theoretical computer…
We study metric Diophantine approximation in local fields of positive characteristic. Specifically, we study the problem of improving Dirichlet's theorem in Diophantine approximation and prove very general results in this context.