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A hypergraph is a generalization of a graph where an edge can join any number of vertices. Unlike typical graphs where edges are pairs of nodes, hyperedges can connect any number of vertices. Example: A set of vertices V = {1, 2, 3, 4} with hyperedges E = {{1, 2}, {2, 3, 4}, {1, 4}}.

In hypergraphs, the degree of a vertex is the number of hyperedges that contain the vertex. Example: In a hypergraph with hyperedges { {1,2,3}, {2,4}, {3,4,5} }, the degree of vertex 3 is 2.

In a hypergraph, a vertex (also called a node) is a fundamental unit that can be connected to other vertices by hyperedges. Example: In a friendship network hypergraph, each person can be represented as a vertex.

A hyperedge is an edge of a hypergraph that can connect any number of vertices. Example: In a hypergraph representing collaborations, a hyperedge could connect all authors of a paper.

Duality in hypergraphs refers to a relationship where the roles of vertices and edges are interchanged. The dual hypergraph of a given hypergraph has a vertex for each hyperedge and a hyperedge for each vertex. Example: If a hypergraph has vertices {1, 2, 3} and hyperedges {{1,2},{1,3}}, its dual has vertices {a, b} and hyperedges {{a,b},{a}}.