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 }, {x ,
1 3 4 5 3 4 1 4 1 2 2
ring => QQ[x ..x ]
1 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 ]
1 4
vertices => {x , x , x , x }
1 2 3 4
------------------------------------------------------------------------
x }}})
4
o2 : Sequence
|
The object randomToricEdgeIdeal is a method function.