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I need to do a matching on a hypergraph. I read that in the case of a hypergraph there is no adjacency matrix.

How do I represent edges then?

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Edges are represented as sets of vertices. With classical graphs, an edge can be represented by the set containing its 2 endpoints. With hypergraphs, they are represented by a set containing more than 2 nodes e.g. $e_i = \lbrace v_1, v_2, ... , v_n \rbrace$.
Incidence matrices are straightforward, just look at the wikipedia page. About incidence matrices you may be interested by this

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