Graph Automorphism Calculator

|Aut(G)| symmetry count

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About Graph Automorphism Calculator

A graph automorphism calculator computing |Aut(G)|: the number of graph symmetries (edge-preserving vertex permutations). |Aut(K_n)|=n!, |Aut(C_n)|=2n, |Aut(P_n)|=2. Vertex/edge-transitive graphs have large automorphism groups. Client-side.

Graph Automorphism Calculator Features

  • |Aut(G)| size
  • Vertex-transitive
  • Edge-transitive
  • Common graphs
  • Orbit-stabilizer
Automorphism: bijection f:V→V preserving edges. Aut(G) forms a group under composition. |Aut(K_n)|=n! (any permutation works). |Aut(C_n)|=2n (n rotations + n reflections = dihedral group). Vertex-transitive: Aut acts transitively on vertices.

How to Use

Select graph:

  • |Aut|: Group size
  • Transitive: Vertex/edge
  • Orbits: Symmetry classes

Transitive Graphs

Vertex-transitive: for any u,v, some automorphism maps u to v. All vertices 'look the same'. Examples: K_n, C_n, Petersen, hypercubes. Edge-transitive: same for edges. Arc-transitive: same for directed edges.

Computing Aut(G)

Practical: nauty (McKay) and bliss algorithms. Theoretical worst case: comparable to graph isomorphism. For most graphs, |Aut(G)|=1 (asymmetric). Random graphs are almost surely asymmetric.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Compute |Aut(G)|.
  3. 3Find orbits.
  4. 4Check transitivity.
  5. 5Identify group.

Graph Automorphism Calculator — Frequently Asked Questions

Are most graphs symmetric?+

No! Almost all graphs are asymmetric: |Aut(G)|=1. The fraction of n-vertex graphs with non-trivial automorphism → 0 as n → ∞. Symmetric graphs are 'special' and well-studied precisely because symmetry is rare.

What's the orbit-stabilizer theorem?+

|Aut(G)| = |orbit(v)| · |stabilizer(v)| for any vertex v. If vertex-transitive: |orbit(v)|=n, so |Aut(G)| = n·|stab(v)|. For K_n: stab(v) = S_{n-1}, so |Aut|=n·(n-1)!=n!.

What's the Petersen graph's automorphism group?+

|Aut(Petersen)|=120=S_5. The Petersen graph is vertex-transitive and edge-transitive. Its automorphism group is isomorphic to S_5 (symmetric group on 5 elements). It's one of the most symmetric graphs.

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