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Jets > jets > jets(ZZ,Graph)

jets(ZZ,Graph) -- the jets of a graph

Synopsis

Description

This function is provided by the package Jets.

Jets of graphs are defined in ยง 2 of F. Galetto, E. Helmick, and M. Walsh, Jet graphs. The input is of type Graph as defined by the EdgeIdeals package, which is automatically exported when loading Jets.

i1 : R= QQ[x,y,z]

o1 = R

o1 : PolynomialRing
i2 : G= graph(R,{{x,y},{y,z}})

o2 = Graph{edges => {{x, y}, {y, z}}}
           ring => R
           vertices => {x, y, z}

o2 : Graph
i3 : JG= jets(2,G)

o3 = Graph{edges => {{x1, y1}, {y1, z1}, {y2, x0}, {y1, x0}, {x2, y0}, {z2, y0}, {x1, y0}, {z1, y0}, {x0, y0}, {y2, z0}, {y1, z0}, {y0, z0}}}
           ring => QQ[x2, y2, z2, x1, y1, z1, x0, y0, z0]
           vertices => {x2, y2, z2, x1, y1, z1, x0, y0, z0}

o3 : Graph
i4 : vertexCovers JG

o4 = {y2*y1*y0, x2*z2*x1*z1*x0*z0, y1*x0*y0*z0, x1*z1*x0*y0*z0}

o4 : List

We can also calculate the jets of a HyperGraph.

i5 : R= QQ[u,v,w,x,y,z]

o5 = R

o5 : PolynomialRing
i6 : H= hyperGraph(R,{{u},{v,w},{x,y,z}})

o6 = HyperGraph{edges => {{u}, {v, w}, {x, y, z}}}
                ring => R
                vertices => {u, v, w, x, y, z}

o6 : HyperGraph
i7 : jets(1,H)

o7 = HyperGraph{edges => {{u1}, {u0}, {w1, v0}, {v1, w0}, {v0, w0}, {z1, x0, y0}, {y1, x0, z0}, {x1, y0, z0}, {x0, y0, z0}}}
                ring => QQ[u1, v1, w1, x1, y1, z1, u0, v0, w0, x0, y0, z0]
                vertices => {u1, v1, w1, x1, y1, z1, u0, v0, w0, x0, y0, z0}

o7 : HyperGraph

Caveat

Rings of jets are usually constructed as towers of rings with tiers corresponding to jets of different orders. However, the tower is flattened out before constructing the edge ideal of the jets of a (hyper)graph. This is done in order to ensure compatibility with the EdgeIdeals package.