The cyclicity of a hypergraph

The cyclicity of a hypergraph is an e ciently computable integer that extends the notion of the cyclomatic number of a graph The formula for the cyclicity is suggested by the join invariant of an acyclic hypergraph which is the multiset of all joining sets in any of its join trees Once we gure out how the multiplicity of a joining set depends on the structure of the acyclic hypergraph we de ne an analogous coe cient the star articulation degree for any subedge of an arbitrary hypergraph and then use it to set up the expression for the cyclicity The basic properties of the cyclicity are that it is zero on acyclic hypergraphs and strictly positive otherwise and that on graphs it coincides with the cyclomatic number we prove these and also some other properties We associate with a hypergraph certain spaces of ows represented by circulant graphs so that the dimension of each space of ows is given by the cyclomatic number of the corresponding circulant graph which is always equal to the cyclicity of the hypergraph We consider one other kind of graphs related to a hypergraph whose cyclomatic number is always equal to the cyclicity of the hypergraph namely join graphs which generalize join trees of an acyclic hypergraph We also compare the cyclicity of a hypergraph with the cyclomatic number of a hypergraph which is like the cyclicity an extension of the cyclomatic number of a graph