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

## Synopsis

• Usage:
(I, G) = randomToricEdgeIdeal(n,t)
• Inputs:
• Outputs:
• I, an ideal, a random toric edge ideal
• G, , the graph underlying I

## 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 }, {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