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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

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