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RandomIdeals :: randomToricEdgeIdeal

randomToricEdgeIdeal -- Creates a toric edge ideal from a random graph with n vertices and t edges.

Synopsis

Description

The toric edge ideal of a graph G is the kernel of the map from the polynomial ring k[edges(G)] to the polynomial ring k[vertices G] taking an xe to yi*yj, where e = (i,j). This method returns the toric edge ideal I of a random graph G which has n vertices and t edges. I is the kernel of the homomorphism from QQ[x1..xn] to QQ/101[e1..et] which sends each vertex in the graph G to the product of its endpoints.

i1 : randomToricEdgeIdeal(4,5)

o1 = (ideal(e e  - e e ), Graph{edges => {{x , x }, {x , x }, {x , x }, {x ,
             1 3    4 5                     3   4     1   4     1   2     2 
                                ring => QQ[x , x , x , x ]
                                            1   2   3   4
                                vertices => {x , x , x , x }
                                              1   2   3   4
     ------------------------------------------------------------------------
     x }, {x , x }}})
      4     1   3

o1 : Sequence

Note that his is different than the randomBinomialEdgeIdeal!

i2 : randomBinomialEdgeIdeal(4,5)

o2 = (ideal (- x y  + x y , - x y  + x y , x y  - x y , x y  - x y , x y  -
                3 4    4 3     3 1    1 3   4 1    1 4   4 2    2 4   1 2  
                                                                           
                                                                           
                                                                           
                                                                           
     ------------------------------------------------------------------------
     x y ), Graph{edges => {{x , x }, {x , x }, {x , x }, {x , x }, {x ,
      2 1                     3   4     1   2     1   3     2   4     1 
                  ring => QQ[x , x , x , x ]
                              1   2   3   4
                  vertices => {x , x , x , x }
                                1   2   3   4
     ------------------------------------------------------------------------
     x }}})
      4

o2 : Sequence

See also

Ways to use randomToricEdgeIdeal :