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- Symmetric_graph abstract "In the mathematical field of graph theory, a graph G is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices u1—v1 and u2—v2 of G, there is an automorphismf : V(G) → V(G)such thatf(u1) = u2 and f(v1) = v2.In other words, a graph is symmetric if its automorphism group acts transitively upon ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1-arc-transitive or flag-transitive.By definition (ignoring u1 and u2), a symmetric graph without isolated vertices must also be vertex transitive. Since the definition above maps one edge to another, a symmetric graph must also be edge transitive. However, an edge-transitive graph need not be symmetric, since a—b might map to c—d, but not to d—c. Semi-symmetric graphs, for example, are edge-transitive and regular, but not vertex-transitive.Every connected symmetric graph must thus be both vertex-transitive and edge-transitive, and the converse is true for graphs of odd degree. However, for even degree, there exist connected graphs which are vertex-transitive and edge-transitive, but not symmetric. Such graphs are called half-transitive. The smallest connected half-transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex-transitive and edge-transitive, rather than an arc-transitive graph. Such a definition would include half-transitive graphs, which are excluded under the definition above.A distance-transitive graph is one where instead of considering pairs of adjacent vertices (i.e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition.A t-arc is defined to be a sequence of t+1 vertices, such that any two consecutive vertices in the sequence are adjacent, and with any repeated vertices being more than 2 steps apart. A t-transitive graph is a graph such that the automorphism group acts transitively on t-arcs, but not on (t+1)-arcs. Since 1-arcs are simply edges, every symmetric graph of degree 3 or more must be t-transitive for some t, and the value of t can be used to further classify symmetric graphs. The cube is 2-transitive, for example.".
- Symmetric_graph thumbnail Petersen1_tiny.svg?width=300.
- Symmetric_graph wikiPageExternalLink foster.
- Symmetric_graph wikiPageExternalLink symmcubic2048list.txt.
- Symmetric_graph wikiPageID "1577896".
- Symmetric_graph wikiPageRevisionID "540806817".
- Symmetric_graph hasPhotoCollection Symmetric_graph.
- Symmetric_graph subject Category:Algebraic_graph_theory.
- Symmetric_graph subject Category:Graph_families.
- Symmetric_graph subject Category:Regular_graphs.
- Symmetric_graph type Abstraction100002137.
- Symmetric_graph type Family108078020.
- Symmetric_graph type GraphFamilies.
- Symmetric_graph type Group100031264.
- Symmetric_graph type Organization108008335.
- Symmetric_graph type SocialGroup107950920.
- Symmetric_graph type Unit108189659.
- Symmetric_graph type YagoLegalActor.
- Symmetric_graph type YagoLegalActorGeo.
- Symmetric_graph type YagoPermanentlyLocatedEntity.
- Symmetric_graph comment "In the mathematical field of graph theory, a graph G is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices u1—v1 and u2—v2 of G, there is an automorphismf : V(G) → V(G)such thatf(u1) = u2 and f(v1) = v2.In other words, a graph is symmetric if its automorphism group acts transitively upon ordered pairs of adjacent vertices (that is, upon edges considered as having a direction).".
- Symmetric_graph label "Grafo simétrico".
- Symmetric_graph label "Graphe symétrique".
- Symmetric_graph label "Symmetric graph".
- Symmetric_graph label "Symmetrischer Graph".
- Symmetric_graph label "Симметричный граф".
- Symmetric_graph label "対称グラフ".
- Symmetric_graph sameAs Symmetrischer_Graph.
- Symmetric_graph sameAs Graphe_symétrique.
- Symmetric_graph sameAs 対称グラフ.
- Symmetric_graph sameAs 대칭_그래프.
- Symmetric_graph sameAs Grafo_simétrico.
- Symmetric_graph sameAs m.05cwvv.
- Symmetric_graph sameAs Q1205074.
- Symmetric_graph sameAs Q1205074.
- Symmetric_graph sameAs Symmetric_graph.
- Symmetric_graph wasDerivedFrom Symmetric_graph?oldid=540806817.
- Symmetric_graph depiction Petersen1_tiny.svg.
- Symmetric_graph isPrimaryTopicOf Symmetric_graph.
