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

Synopsis

Function: jets
Usage:

jets (n,G)
Inputs:
- n, an integer,
- G, a graph, undirected, finite, and simple graph or hypergraph
Optional inputs:
- Projective => ..., default value false, Option for jets
Outputs:
- a graph, the (hyper)graph of n-jets of G

Description

This function is provided by the package Jets.

Jets of graphs are defined in § 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.

Ways to use this method: