Related papers: A coalgebraic model of graphs
The paper consider an equivalence relation in the set of vertices of a bipartite graph. Some numerical characteristics showing the cardinality of equivalence classes are introduced. A combinatorial identity that is in relationship to these…
This paper is mainly a semi-tutorial introduction to elementary algebraic topology and its applications to Ising-type models of statistical physics, using graphical models of linear and group codes. It contains new material on systematic…
In this article, we combine Sweedler's classic theory of measuring coalgebras -- by which $k$-algebras are enriched in $k$-coalgebras for $k$ a field -- with the theory of W-types -- by which the categorical semantics of inductive data…
Let $G_{1}$ and $G_{2}$ be disjoint copies of a graph $G$, and let $g:V(G_{1})\rightarrow V(G_{2})$ be a function. A functigraph $F_{G}$ consists of the vertex set $V(G_{1})\cup V(G_{2})$ and the edge set $E(G_{1})\cup E(G_{2})\cup…
In previous work, categories of algebras of endofunctors were shown to be enriched in categories of coalgebras of the same endofunctor, and the extra structure of that enrichment was used to define a generalization of inductive data types.…
We explain how categories, and groupoids, can be seen as models for a Lawvere ${\mathfrak Gr}$-theory, where ${\mathfrak Gr}$ is the category of graphs, and show that for Lawvere ${\mathfrak Gr}$-theories finitely presentable models are…
In this paper, we study the graph classification problem from the graph homomorphism perspective. We consider the homomorphisms from $F$ to $G$, where $G$ is a graph of interest (e.g. molecules or social networks) and $F$ belongs to some…
We propose the use of hyperedge replacement graph grammars for factor graphs, or factor graph grammars (FGGs) for short. FGGs generate sets of factor graphs and can describe a more general class of models than plate notation, dynamic…
Any simple group-grading of a finite dimensional complex algebra induces a natural family of digraphs. We prove that $|E\circ E^{\text{op}}\cup E^{\text{op}}\circ E|\geq |E|$ for any digraph $\Gamma =(V,E)$ without parallel edges, and…
In this paper we introduce a class of hypergraphs that we call chordal. We also extend the definition of triangulated hypergraphs, given in \cite{VT}, so that a triangulated hypergraph, according to our definition, is a natural…
In the theory of coalgebras $C$ over a ring $R$, the rational functor relates the category of modules over the algebra $C^*$ (with convolution product) with the category of comodules over $C$. It is based on the pairing of the algebra $C^*$…
We investigate models of algebraic theories in the category of cocommutative coalgebras over a field. We establish some of their categorical properties, similar to those of algebraic varieties. We introduce a class of categories of…
We introduce a class of algebras over a field $\mathbb{F}$ related to directed graphs in which all edges are labeled by nonzero elements of the field $\mathbb{F}$. If all labels are different from $1$, these algebras are axial algebras. We…
We introduce the notion of a diagram category and discuss its application to the invariant theory of classical groups and super groups, with some indications concerning extensions to quantum groups and quantum super groups. Tensor functors…
The language of graph theory, or network science, has proven to be an exceptional tool for addressing myriad problems in neuroscience. Yet, the use of networks is predicated on a critical simplifying assumption: that the quintessential unit…
A graph with a semiregular group of automorphisms can be thought of as the derived cover arising from a voltage graph. Since its inception, the theory of voltage graphs and their derived covers has been a powerful tool used in the study of…
We continue the investigation of tabular algebras with trace (a certain class of associative ${\Bbb Z}[v, v^{-1}]$-algebras equipped with distinguished bases) by determining the extent to which the tabular structure may be recovered from a…
The algebra of eikonals $\mathfrak E$ of a metric graph $\Omega$ is an operator $C^*$-algebra determined by dynamical system with boundary control that describes wave propagation on the graph. In this paper, two canonical block forms…
The concept of viewing graph solvability has gained significant interest in the context of structure-from-motion. A viewing graph is a mathematical structure where nodes are associated to cameras and edges represent the epipolar geometry…
A graph is a data structure composed of dots (i.e. vertices) and lines (i.e. edges). The dots and lines of a graph can be organized into intricate arrangements. The ability for a graph to denote objects and their relationships to one…