1 | initial version |

You have first to create your polynomial ring with variables names being the vertices of your graph. But since some vertices are numbers, and since the variable names must atart with a letter, i propose to name them `x_i`

where `i`

is the string representing each vertex:

```
sage: G = graphs.PetersenGraph()
sage: R = PolynomialRing(QQ,['x_'+str(i) for i in G.vertices()])
sage: R
Multivariate Polynomial Ring in x_0, x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9 over Rational Field
sage: R.inject_variables()
Defining x_0, x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9
```

Then you can define the list of polynomials that define your ideal:

```
sage: L = [R('x_'+str(i)) * R('x_'+str(j)) for i,j in G.edges(labels=False)]
```

Then you can construct your ideal:

```
sage: I = R.ideal(L)
sage: I
Ideal (x_0*x_1, x_0*x_4, x_0*x_5, x_1*x_2, x_1*x_6, x_2*x_3, x_2*x_7, x_3*x_4, x_3*x_8, x_4*x_9, x_5*x_7, x_5*x_8, x_6*x_8, x_6*x_9, x_7*x_9) of Multivariate Polynomial Ring in x_0, x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9 over Rational Field
```

2 | No.2 Revision |

You have first to create your polynomial ring with variables names being the vertices of your graph. But since some vertices are ~~numbers, ~~usually numbers (thougt they can be other Sage objects), and since the variable names must ~~atart ~~start with a letter, i propose to name them `x_i`

where `i`

is the string representing each vertex:

```
sage: G = graphs.PetersenGraph()
sage: R = PolynomialRing(QQ,['x_'+str(i) for i in G.vertices()])
sage: R
Multivariate Polynomial Ring in x_0, x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9 over Rational Field
sage: R.inject_variables()
Defining x_0, x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9
```

Then you can define the list of polynomials that define your ideal:

```
sage: L = [R('x_'+str(i)) * R('x_'+str(j)) for i,j in G.edges(labels=False)]
```

Then you can construct your ideal:

```
sage: I = R.ideal(L)
sage: I
Ideal (x_0*x_1, x_0*x_4, x_0*x_5, x_1*x_2, x_1*x_6, x_2*x_3, x_2*x_7, x_3*x_4, x_3*x_8, x_4*x_9, x_5*x_7, x_5*x_8, x_6*x_8, x_6*x_9, x_7*x_9) of Multivariate Polynomial Ring in x_0, x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9 over Rational Field
```

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