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A relational structure $\mathbb{X}$ is called reversible iff each bijective homomorphism from $\mathbb{X}$ onto $\mathbb{X}$ is an isomorphism, and linear orders are prototypical examples of such structures. One way to detect new reversible…
Given a system $\mathcal{G} =\{G_1,G_2,\dots,G_m\}$ of graphs/digraphs/hypergraphs on the common vertex set $V$ of size $n$, an $m$-edge graph/digraph/hypergraph $H$ on $V$ is transversal in $\mathcal{G}$ if there exists a bijection…
The representation of complex systems as networks is inappropriate for the study of certain problems. We show several examples of social, biological, ecological and technological systems where the use of complex networks gives very limited…
In this paper, we count acyclic and strongly connected uniform directed labeled hypergraphs. For these combinatorial structures, we introduce a specific generating function allowing us to recover and generalize some results on the number of…
In this dissertation, we explore the structure of inversion graphs of permutations--a class of graphs that naturally arises by representing each permutation as a graph, where vertices correspond to entries and edges encode inversions.…
Hypergraphs, increasingly utilised to model complex and diverse relationships in modern networks, have gained significant attention for representing intricate higher-order interactions. Among various challenges, cohesive subgraph discovery…
Hypertrees are high-dimensional counterparts of graph theoretic trees. They have attracted a great deal of attention by various investigators. Here we introduce and study Hyperpaths -- a particular class of hypertrees which are high…
As data structures and mathematical objects used for complex systems modeling, hypergraphs sit nicely poised between on the one hand the world of network models, and on the other that of higher-order mathematical abstractions from algebra,…
Algebraic hyperstructures represent a natural extension of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two…
A graph is a structure composed of a set of vertices (i.e.nodes, dots) connected to one another by a set of edges (i.e.links, lines). The concept of a graph has been around since the late 19$^\text{th}$ century, however, only in recent…
Recursive graph queries are increasingly popular for extracting information from interconnected data found in various domains such as social networks, life sciences, and business analytics. Graph data often come with schema information that…
In this paper we show how the hyperstructure concept leads to new algebraic structures and general field theories.
I introduce a class of diagrams in a Grothendieck site called "atlases" which can be used to study hyperdescent, and show that hypersheaves take atlases to limits using an indexed `nerve' construction that produces hypercovers from atlases.…
Graphs provide a natural representation of relational structure that arises across diverse domains. Despite this ubiquity, graph structure is typically learned in a modality- and task-isolated manner, where graph representations are…
In this paper we investigate a ternary generalization of associativity by defining a diagrammatic calculus of hypergraphs that extends the usual notions of tensor networks, categories and relational algebras. In doing so we rediscover the…
Recently, deep architectures, such as recurrent and recursive neural networks have been successfully applied to various natural language processing tasks. Inspired by bidirectional recurrent neural networks which use representations that…
An oriented hypergraph is an oriented incidence structure that extends the concept of a signed graph. We introduce hypergraphic structures and techniques central to the extension of the circuit classification of signed graphs to oriented…
We define recurrence matrices and study a few properties (links with automatic sequences, branch groups etc.) of them.
We present a new and powerful algebraic framework for graph rewriting, based on drags, a class of graphs enjoying a novel composition operator. Graphs are embellished with roots and sprouts, which can be wired together to form edges. Drags…
We study the first order theory of structures over graphs i.e. structures of the form ($\mathcal{G},\tau$) where $\mathcal{G}$ is the set of all (isomorphism types of) finite undirected graphs and $\tau$ some vocabulary. We define the…