|
|
|
|
Graph automorphism
|
| |
|
| |
In graph-theoretical mathematics, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity.
Formally, an automorphism of a graph G = (V,E) is a permutation σ of the vertex set V, such that for any edge e = (u,v), σ(e) = (σ(u),σ(v)) is also an edge.
Discussion
Ask a question about 'Graph automorphism' Start a new discussion about 'Graph automorphism' Answer questions from other users |
Encyclopedia
In graph-theoretical mathematics, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity.
Formally, an automorphism of a graph G = (V,E) is a permutation σ of the vertex set V, such that for any edge e = (u,v), σ(e) = (σ(u),σ(v)) is also an edge. That is, it is a graph isomorphism from G to itself. Automorphisms may be defined in this way both for directed graphs and for undirected graphs.
The composition of two automorphisms is another automorphism, and the set of automorphisms of a given graph, under the composition operation, forms a group, the automorphism group of the graph.
Asymmetric graphs
The identity mapping of a graph onto itself is called the trivial automorphism of the graph. An asymmetric graph is a graph which has only the trivial automorphism.
The smallest asymmetric non-trivial graphs have 6 vertices.
The proportion of graphs on n vertices with nontrivial automorphism tends to zero as n grows, which is informally expressed as "almost all graphs are asymmetric".
Computational complexity
Constructing the automorphism group is at least as difficult (in terms of its computational complexity) as solving the graph isomorphism problem, determining whether two given graphs correspond vertex-for-vertex and edge-for-edge. For, G and H are isomorphic if and only if the disconnected graph formed by the disjoint union of graphs G and H has an automorphism that swaps the two components.
The graph automorphism problem is the problem of testing whether a graph has a nontrivial automorphism. It belongs to the class NP of computational complexity. Similar to the graph isomorphism problem, it is unknown whether it has a polynomial time algorithm or it is NP-complete. It is known that the graph automorphism problem is polynomial-time many-one reducible to graph isomorphism problem, but the converse reduction is unknown.
Symmetry display
Several graph drawing researchers have investigated algorithms for drawing graphs in such a way that the automorphisms of the graph become visible as symmetries of the drawing. This may be done either by using a method that is not designed around symmetries, but that automatically generates symmetric drawings when possible, or by explicitly identifying symmetries and using them to guide vertex placement in the drawing. It is not always possible to display all symmetries of the graph simultaneously, so it may be necessary to choose which symmetries to display and which to leave unvisualized.
Graph families defined by their automorphisms
Several families of graphs are defined by having certain types of automorphisms.
- A vertex-transitive graph is an undirected graph in which, for every pair of vertices u and v, there is an automorphism mapping u to v.
- An edge-transitive graph is an undirected graph in which, for every pair of edges e and f, there is an automorphism mapping e to f.
- A symmetric graph is a graph that is both vertex-transitive and edge-transitive.
- An arc-transitive graph is an undirected graph, in which any pair of a vertex u and an edge (u,v) may be mapped by an automorphism to any other such pair.
- A distance-transitive graph is a graph such that every pair of vertices may be mapped by an automorphism into any other pair of vertices that are the same distance apart.
- A semi-symmetric graph is a graph that is edge-transitive but not vertex-transitive.
- A skew-symmetric graph is a directed graph together with a permutation σ on the vertices that maps edges to edges but reverses the direction of each edge. Additionally, σ is required to be an involution.
|
| |
|
|
