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WOLFRAM|DEMONSTRATIONS PROJECT

Simple Graphs and Their Binomial Edge Ideals

​
first vertex
0
1
2
3
4
5
6
7
8
second vertex
0
1
2
3
4
5
6
7
8
add/delete vertex
0
1
2
3
4
5
6
7
8
add edge
delete edge
add vertex
delete vertex
new random graph
complete graph
maximum degreee is: 7
Groebner Basis is:
{
x
6
y
1
y
2
y
3
y
4
y
5
y
8
-
x
8
y
1
y
2
y
3
y
4
y
5
y
6
,
x
5
y
2
y
3
y
4
y
8
-
x
8
y
2
y
3
y
4
y
5
,
x
4
y
3
y
8
-
x
8
y
3
y
4
,
x
3
y
8
-
x
8
y
3
,
x
5
y
1
y
6
-
x
6
y
1
y
5
,
x
1
y
6
-
x
6
y
1
,
x
4
y
2
y
5
-
x
5
y
2
y
4
,
x
2
y
5
-
x
5
y
2
,
x
1
y
5
-
x
5
y
1
,
x
3
y
4
-
x
4
y
3
,
x
2
y
4
-
x
4
y
2
,
x
2
x
4
y
3
-
x
3
x
4
y
2
,
x
1
x
5
y
2
-
x
2
x
5
y
1
}
This Demonstration illustrates the relationship between combinatorial properties of a simple graph and its binomial edge ideal (see the Details section for definitions). In particular, it can be used to verify that a graph is closed (for a given ordering of vertices) if and only if the Groebner basis of its edge ideal consists of quadratic polynomials. By starting with a random graph that is not closed and adding suitable edges until the Groebner basis consists only of quadratic polynomials, you can find the closure of the graph, that is, the minimal closed graph containing the given graph. Alternatively, you can start with a complete graph (which is always closed) and remove edges (or vertices) to obtain non-closed graphs.
To add/delete a vertex, choose the vertex number from the third setter bar. To add/delete an edge, choose the first and second vertex of the edge from the first two setter bars.
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