Convert graph into ideal in polynomial ring

graph
ideal

asked 2016-01-12 12:12:15 +0100

selva
45 ●2 ●7

Let $G = (V (G), E(G))$ denote a finite simple (no loops or multiple edges) undirected graph with vertices $V (G) =\ {x_1 ,\ldots, x_n }$ and edge set $E(G)$ . By identifying the vertices with the variables in the polynomial ring $R = k[x_1 ,\ldots, x_n ]$ (where $k$ is a field), we can associate to each simple graph $G$ a monomial ideal $ I(G) = ({ x_i x_j |{x_i , x_j } \in E(G)})$

How to convert graph into ideal in sage ? Suppose $G$ is cycle graph. can i get ideal with generator $(x_1x_2,x_2x_3,x_3x_4,x_4x_5,x_1x_5)$ in $k[x_1,\ldots ,x_5]$ Please give some hint.

Thanks in advance

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answered 2016-01-12 13:50:53 +0100

tmonteil
27423 ●31 ●204 ●517 http://wiki.sagemath.o...

updated 2016-01-12 13:52:58 +0100

You have first to create your polynomial ring with variables names being the vertices of your graph. But since some vertices are usually numbers (thougt they can be other Sage objects), and since the variable names must 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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Great answer! But how do we convert the list L to a polynomial object in the ring R?

vidyarthi ( 2023-03-13 16:09:18 +0100 )edit

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answered 2016-01-12 19:41:14 +0100

slelievre
17704 ●22 ●160 ●350 http://carva.org/samue...

You could do the following.

sage: k = QQ
sage: n = 10
sage: R = PolynomialRing(k, 'x', n)
sage: R
Multivariate Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7, x8, x9 over Rational Field
sage: R.inject_variables()
Defining x0, x1, x2, x3, x4, x5, x6, x7, x8, x9
sage: verts = R.gens()
sage: edges = [(x1, x2), (x3, x7), (x4, x9)]
sage: G = Graph([verts, edges], format='vertices_and_edges')
sage: G
Graph on 10 vertices
sage: J = R.ideal([a * b for a, b in G.edges(labels=False)])
sage: J
Ideal (x4*x9, x3*x7, x1*x2) of Multivariate Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7, x8, x9 over Rational Field

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