# hyperGraphToSimplicialComplex -- makes a simplicial complex from a (hyper)graph

## Synopsis

• Usage:
D = hyperGraphToSimplicialComplex H
• Inputs:
• H, ,
• Outputs:
• D, , whose facets are given by the edges of H

## Description

This function produces a simplicial complex from a (hyper)graph. The facets of the simplicial complex are given by the edge set of the (hyper)graph. This function is the inverse of simplicialComplexToHyperGraph and enables users to make use of functions in the package SimplicialComplexes.

 i1 : R = QQ[x_1..x_6]; i2 : G = graph({x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_5,x_1*x_5,x_1*x_6,x_5*x_6}) --5-cycle and a triangle o2 = Graph{edges => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }}} 1 2 2 3 3 4 1 5 4 5 1 6 5 6 ring => R vertices => {x , x , x , x , x , x } 1 2 3 4 5 6 o2 : Graph i3 : DeltaG = hyperGraphToSimplicialComplex G o3 = simplicialComplex | x_5x_6 x_1x_6 x_4x_5 x_1x_5 x_3x_4 x_2x_3 x_1x_2 | o3 : SimplicialComplex i4 : hyperGraphDeltaG = simplicialComplexToHyperGraph DeltaG o4 = HyperGraph{edges => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }}} 1 2 2 3 3 4 1 5 4 5 1 6 5 6 ring => R vertices => {x , x , x , x , x , x } 1 2 3 4 5 6 o4 : HyperGraph i5 : GPrime = graph(hyperGraphDeltaG) o5 = Graph{edges => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }}} 1 2 2 3 3 4 1 5 4 5 1 6 5 6 ring => R vertices => {x , x , x , x , x , x } 1 2 3 4 5 6 o5 : Graph i6 : G === GPrime o6 = true