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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 x_e to y_i*y_j, 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[x_1..x_n] to QQ/101[e_1..e_t] 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 },
             1 3    4 5                       3   4     1   4     1   2  
                                "ring" => QQ[x ..x ]
                                              1   4
                                "vertices" => {x , x , x , x }
                                                1   2   3   4
     ------------------------------------------------------------------------
     {x , x }, {x , x }}})
       2   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 ]
                                1   4
                  "vertices" => {x , x , x , x }
                                  1   2   3   4
     ------------------------------------------------------------------------
     x }}})
      4

o2 : Sequence

See also

Ways to use randomToricEdgeIdeal :

For the programmer

The object randomToricEdgeIdeal is a method function.