Symmetric graph

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 automorphism

f : V(G) → V(G)
The Petersen graph is a (cubic) symmetric graph. Any pair of adjacent vertices can be mapped to another by an automorphism, since any five-vertex ring can be mapped to any other.

such that

f(u1) = u2 and f(v1) = v2.[1]

In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction).[2] Such a graph is sometimes also called 1-arc-transitive[2] or flag-transitive.[3]

By definition (ignoring u1 and u2), a symmetric graph without isolated vertices must also be vertex-transitive.[1] 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. Star graphs are a simple example of being edge-transitive without being vertex-transitive or symmetric. As a further example, semi-symmetric graphs are edge-transitive and regular, but not vertex-transitive.

Graph families defined by their automorphisms
distance-transitive → distance-regular ← strongly regular
↓
symmetric (arc-transitive) ← t-transitive, t ≥ 2 skew-symmetric
↓
(if connected)
vertex- and edge-transitive		 → edge-transitive and regular → edge-transitive
↓ ↓ ↓
vertex-transitive → regular → (if bipartite)
biregular
↑
Cayley graph ← zero-symmetric asymmetric

Every connected symmetric graph must thus be both vertex-transitive and edge-transitive, and the converse is true for graphs of odd degree.[3] However, for even degree, there exist connected graphs which are vertex-transitive and edge-transitive, but not symmetric.[4] Such graphs are called half-transitive.[5] The smallest connected half-transitive graph is Holt's graph, with degree 4 and 27 vertices.[1][6] 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.[1]

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.[1]

Examples

Combining the symmetry condition with the restriction that graphs be cubic (i.e. all vertices have degree 3) yields quite a strong condition, and such graphs are rare enough to be listed. The Foster census and its extensions provide such lists.[7] The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs,[8] and in 1988 (when Foster was 92[1]) the then current Foster census (listing all cubic symmetric graphs up to 512 vertices) was published in book form.[9] The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices[10][11] (ten of these are also distance-transitive; the exceptions are as indicated):

VerticesDiameterGirthGraphNotes
413The complete graph K4distance-transitive, 2-arc-transitive
624The complete bipartite graph K3,3distance-transitive, 3-arc-transitive
834The vertices and edges of the cubedistance-transitive, 2-arc-transitive
1025The Petersen graphdistance-transitive, 3-arc-transitive
1436The Heawood graphdistance-transitive, 4-arc-transitive
1646The Möbius–Kantor graph2-arc-transitive
1846The Pappus graphdistance-transitive, 3-arc-transitive
2055The vertices and edges of the dodecahedrondistance-transitive, 2-arc-transitive
2056The Desargues graphdistance-transitive, 3-arc-transitive
2446The Nauru graph (the generalized Petersen graph G(12,5))2-arc-transitive
2656The F26A graph[11]1-arc-transitive
2847The Coxeter graphdistance-transitive, 3-arc-transitive
3048The Tutte–Coxeter graphdistance-transitive, 5-arc-transitive

Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance-transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance-transitive graphs.

Non-cubic symmetric graphs include cycle graphs (of degree 2), complete graphs (of degree 4 or more when there are 5 or more vertices), hypercube graphs (of degree 4 or more when there are 16 or more vertices), and the graphs formed by the vertices and edges of the octahedron, icosahedron, cuboctahedron, and icosidodecahedron. The Rado graph forms an example of a symmetric graph with infinitely many vertices and infinite degree.

Properties

The vertex-connectivity of a symmetric graph is always equal to the degree d.[3] In contrast, for vertex-transitive graphs in general, the vertex-connectivity is bounded below by 2(d + 1)/3.[2]

A t-transitive graph of degree 3 or more has girth at least 2(t – 1). However, there are no finite t-transitive graphs of degree 3 or more for t ≥ 8. In the case of the degree being exactly 3 (cubic symmetric graphs), there are none for t ≥ 6.

See also

References
  1. Biggs, Norman (1993). Algebraic Graph Theory (2nd ed.). Cambridge: Cambridge University Press. pp. 118–140. ISBN 0-521-45897-8.
  2. Godsil, Chris; Royle, Gordon (2001). Algebraic Graph Theory. New York: Springer. p. 59. ISBN 0-387-95220-9.
  3. Babai, L (1996). "Automorphism groups, isomorphism, reconstruction". In Graham, R; Grötschel, M; Lovász, L (eds.). Handbook of Combinatorics. Elsevier.
  4. Bouwer, Z. "Vertex and Edge Transitive, But Not 1-Transitive Graphs." Canad. Math. Bull. 13, 231–237, 1970.
  5. Gross, J.L. & Yellen, J. (2004). Handbook of Graph Theory. CRC Press. p. 491. ISBN 1-58488-090-2.
  6. Holt, Derek F. (1981). "A graph which is edge transitive but not arc transitive". Journal of Graph Theory. 5 (2): 201–204. doi:10.1002/jgt.3190050210..
  7. Marston Conder, Trivalent symmetric graphs on up to 768 vertices, J. Combin. Math. Combin. Comput, vol. 20, pp. 41–63
  8. Foster, R. M. "Geometrical Circuits of Electrical Networks." Transactions of the American Institute of Electrical Engineers 51, 309–317, 1932.
  9. "The Foster Census: R.M. Foster's Census of Connected Symmetric Trivalent Graphs", by Ronald M. Foster, I.Z. Bouwer, W.W. Chernoff, B. Monson and Z. Star (1988) ISBN 0-919611-19-2
  10. Biggs, p. 148
  11. Weisstein, Eric W., "Cubic Symmetric Graph", from Wolfram MathWorld.
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