Graphs

Symbolica documentation for getting started, symbolic expressions, numerical evaluation, pattern matching, and APIs in Python and Rust.

Run Symbolica in your browser

Python examples with a Run button can be edited and run here. The first run loads Python, Symbolica, hover documentation, and code completion.

Graphs with parallel edges, self-edges, edge directions, vertex and edge colors can easily be constructed. For example:

from symbolica import *

g = Graph()
n1 = g.add_node(4)
n2 = g.add_node(5)
g.add_edge(n1, n2)
g.add_edge(n2, n1, True, E("x^2+5"))

g

Run the example to draw the graph. Edit its nodes or edges and run it again to update the drawing.

graph TD;
  0[4];
  1[5];
  0 --- 1;
  1 -->|x^2+5| 0;

The data of the nodes an edges can be any data in Rust and in Python it is a Symbolica Expression.

Export

Graphs can be written in Graphviz “dot” notation or in mermaid notation. For example:

print(g.to_dot())

Output

digraph G {
  0 [label="4"];
  1 [label="5"];
  0 -> 1 [dir=none,label="0"];
  1 -> 0 [label="5+x^2"];
}

Isomorphisms

You can check if two graphs are isomorphic using is_isomorphic.

Canonization

A graph can be written into its canonical form: a specific representation that any isomorphic version of the graph will map to.

For example:

from symbolica import *

g = Graph()
n0 = g.add_node(1)
n1 = g.add_node(0)
n2 = g.add_node(1)
n3 = g.add_node(0)
n4 = g.add_node(2)
n5 = g.add_node(0)
n6 = g.add_node(1)
n7 = g.add_node(0)
n8 = g.add_node(1)
g.add_edge(n0, n1)
g.add_edge(n0, n3)
g.add_edge(n1, n2)
g.add_edge(n1, n3)
g.add_edge(n1, n4)
g.add_edge(n1, n5)
g.add_edge(n2, n5)
g.add_edge(n3, n4)
g.add_edge(n3, n6)
g.add_edge(n3, n7)
g.add_edge(n4, n5)
g.add_edge(n4, n7)
g.add_edge(n5, n7)
g.add_edge(n5, n8)
g.add_edge(n6, n7)
g.add_edge(n7, n8)
canonical, vertex_map, group_size, orbits = g.canonize()
print("Vertex map:", vertex_map)
print("Automorphism group size:", group_size)
print("Orbits:", orbits)
canonical

graph TD;
  0[0];
  1[1];
  2[2];
  3[3];
  4[4];
  5[5];
  6[6];
  7[7];
  8[8];
  0 --- 2;
  0 --- 3;
  0 --- 6;
  0 --- 7;
  0 --- 8;
  1 --- 2;
  1 --- 3;
  1 --- 4;
  1 --- 5;
  1 --- 8;
  2 --- 5;
  2 --- 7;
  2 --- 8;
  3 --- 4;
  3 --- 6;
  3 --- 8;

Returns the canonized graph, the vertex map ([7, 0, 6, 2, 8, 3, 5, 1, 4]), the automorphism group size (8), and the orbit ((0 1 2 3)(4 5 6 7)(8)).

Graph generation

Symbolica can generate all connected graphs with external_edges half-edges and the given allowed list of vertex connections. The generator returns the canonical form of the graph and the size of its automorphism group (including edge permutations).

For example:

from symbolica import *

g, q, gh = HalfEdge(S("g")), HalfEdge(S("q"), True), HalfEdge(S("gh"), True)

graphs = Graph.generate(
    [(1, g), (2, g)],
    [[g, g, g], [g, g, g, g], [q.flip(), q, g], [gh.flip(), gh, g]],
    max_loops=2,
)

[(f'Symmetry factor = 1/{sym}', g) for g, sym in graphs.items()]

This generates all connected Feynman graphs for QCD up to 2 loops with the specified vertices. The list of caption-and-graph pairs is displayed as a gallery with the symmetry factors.