Related papers: Minimally Connected Hypergraphs
Hypergraphs, encoding structured interactions among any number of system units, have recently proven a successful tool to describe many real-world biological and social networks. Here we propose a framework based on statistical inference to…
Hypergraph data, which capture multi-way interactions among entities, are increasingly prevalent in the big data era. Generating new hyperlinks from an observed, usually high-dimensional hypergraph is an important yet challenging task with…
In this paper we focus on the problem of finding (small) subhypergraphs in a (large) hypergraph. We use this problem to illustrate that reducing hypergraph problems to graph problems by working with the 2-section is not always a reasonable…
Random hypergraphs extend the classical notion of random graphs by allowing hyperedges to join more than two vertices, making them well-suited for modeling higher-order interactions in complex systems. Despite their broad applicability,…
One of the defining features of complex networks is the connectivity properties that we observe emerging from local interactions. Recently, hypergraphs have emerged as a versatile tool to model networks with non-dyadic, higher-order…
Directed graphs are widely used in modelling of nonsymmetric relations in various sciences and engineering disciplines. We discuss invariants of strongly connected directed graphs - minimal number of vertices or edges necessary to remove to…
We derive an asymptotic formula for the number of connected 3-uniform hypergraphs with vertex set $[N]$ and $M$ edges for $M=N/2+R$ as long as $R$ satisfies $R = o(N)$ and $R=\omega(N^{1/3}\ln^{2} N)$. This almost completely fills the gap…
A network can be analyzed at different topological scales, ranging from single nodes to motifs, communities, up to the complete structure. We propose a novel intermediate-level topological analysis that considers non-overlapping subgraphs…
A subset $M$ of the edges of a graph or hypergraph is hitting if $M$ covers each vertex of $H$ at least once, and $M$ is $t$-shallow if it covers each vertex of $H$ at most $t$ times. We consider the existence of shallow hitting edge sets…
Graph invariants provide a powerful analytical tool for investigation of abstract structures of graphs. They, combined in convenient relations, carry global and general information about a graph and its various substructures such as cycle…
We describe the structure of connected graphs with the minimum and maximum average distance, radius, diameter, betweenness centrality, efficiency and resistance distance, given their order and size. We find tight bounds on these graph…
We generalize a result of Balister, Gy{\H{o}}ri, Lehel and Schelp for hypergraphs. We determine the unique extremal structure of an $n$-vertex, $r$-uniform, connected, hypergraph with the maximum number of hyperedges, without a…
In this survey we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by…
We propose a tensor product structure that is compatible with the hypergraph structure. We define the algebraic connectivity of the $(m+1)$-uniform hypergraph in this product, and prove the relationship with the vertex connectivity. We…
The notion of graph covers is a discretization of covering spaces introduced and deeply studied in topology. In discrete mathematics and theoretical computer science, they have attained a lot of attention from both the structural and…
We introduce adequate concepts of expansion of a digraph to obtain a sequential construction of minimal strong digraphs. We characterize the class of minimal strong digraphs whose expansion preserves the property of minimality. We prove…
Let $(X,\mathcal{E})$ be a hypergraph. A support is a graph $Q$ on $X$ such that for each $E\in\mathcal{E}$, the subgraph of $Q$ induced on the elements in $E$ is connected. We consider the problem of constructing a support for hypergraphs…
Counting small patterns in a large dataset is a fundamental algorithmic task. The most common version of this task is subgraph/homomorphism counting, wherein we count the number of occurrences of a small pattern graph $H$ in an input graph…
Here we study the spectral properties of an underlying weighted graph of a non-uniform hypergraph by introducing different connectivity matrices, such as adjacency, Laplacian and normalized Laplacian matrices. We show that different…
A constructive characterization of the class of uniformly $4$-connected graphs is presented. The characterization is based on the application of graph operations to appropriate vertex and edge sets in uniformly $4$-connected graphs, that…