Related papers: Graphs, hypergraphs, and properads
We give an algebraic presentation of directed acyclic graph structure, introducing a symmetric monoidal equational theory whose free PROP we characterise as that of finite abstract dags with input/output interfaces. Our development provides…
We describe proper actors from the underlying groupoid of a graph C*-algebra to another \'etale groupoid in terms of bisections. This allows to understand graph morphisms and the *-homomorphisms that they induce more conceptually. More…
Graph monoids arise naturally in the study of non-stable K-theory of graph C*-algebras and Leavitt path algebras. They play also an important role in the current approaches to the realization problem for von Neumann regular rings. In this…
By natural way the hierarchy structure is introduced on directed graphs with weighted adjacencies. Embedded system of algebras of subsets of the set of vertices of such digraph and it's consolidations, which vertices are the elementary sets…
We introduce a taxonomy of interaction types and show that graphs are focal hypergraphs: every graph is canonically a focal hypergraph via its closed neighbourhood structure, and every graph dynamical model is a special case of the general…
Notions of freedom and independence for hypergraphs of models of a theory are defined. Properties of these notions and their applications to some natural classes of theories are studied.
We introduce a homotopy theory of digraphs (directed graphs) and prove its basic properties, including the relations to the homology theory of digraphs constructed by the authors in previous papers. In particular, we prove the homotopy…
We extend the basic concepts of Street's formal theory of monads from the setting of 2-categories to that of double categories. In particular, we introduce the double category Mnd(C) of monads in a double category C and define what it means…
The notions of spectral measures and spectral classes, which are well known for graphs, are generalized and investigated for oriented hypergraphs.
Many real-world phenomena are naturally modeled by graphs and networks. However, classical graph models are often limited to pairwise interactions and may not adequately capture the richer structures that arise in practice. Higher-order…
We describe various path homology theories constructed for a directed hypergraph. We introduce the category of directed hypergraphs and the notion of a homotopy in this category. Also, we investigate the functoriality and the homotopy…
We characterise the structure of those graphs of a given order which maximise the number of connected induced subgraphs for seven different graph classes, each with other prescribed parameters like minimum degree, independence number,…
We develop a theory of infinity properads enriched in a general symmetric monoidal infinity category. These are defined as presheaves, satisfying a Segal condition and a Rezk completeness condition, over certain categories of graphs. In…
A self-contained exposition is given of the topological and Galois-theoretic properties of the category of combinatorial 1-complexes, or graphs, very much in the spirit of Stallings. A number of classical, as well as some new results about…
We present the basic ideas of forms (a generalization of Ehresmann's sketches) and their theories and models, more explicitly than in previous expositions. Forms provide the ability to specify mathematical structures and data types in any…
The theory of finite automata concerns itself with words in a free monoid together with concatenation and without further structure. There are, however, important applications which use alphabets which are structured in some sense. We…
The concept of n-categories and related subject is considered. An n-category is described as an n-graph with a composition. A new definition of operad is presented. Some illustrative examples are given.
Graphs are commonly used to characterise interactions between objects of interest. Because they are based on a straightforward formalism, they are used in many scientific fields from computer science to historical sciences. In this paper,…
Based on the monoid classifier, we give an alternative axiomatization of Freyd's paracategories, which can be interpreted in any bicategory of partial maps. Assuming furthermore a free-monoid monad T in our ambient category, and…
We demonstrate that graph-based models are fully capable of representing higher-order interactions, and have a long history of being used for precisely this purpose. This stands in contrast to a common claim in the recent literature on…