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Automorphic Graph

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An automorphic graph is a distance-transitive graph Gamma for which the automorphism group Aut(Gamma) acts primitively on the vertices of Gamma and is not a complete graph or an line graph (Biggs 1993, p. 178).

The following table summarizes some known automorphic graphs (Gordon and Levingston 1981; Biggs 1976; Biggs 1993, pp. 178-179). There are exactly three cubic automorphic graphs, and a single quartic automorphic graph (Biggs 1976). Note that odd graphs are automorphic (Biggs 1976).

|Gamma|degreegraphintersection array
103Petersen graph{3,2;1,1}
283Coxeter graph{3,2,2,1;1,1,1,2}
354odd graph O_4{4,3,3;1,1,2}
36Sylvester graph{5,4,2;1,1,4}
57Perkel graph{6,5,2;1,1,3}
63Conway-Smith graph{6,4,4;1,1,3}
65Hall graph{10,6,4;1,2,5}
68Doro graph{12,10,3;1,3,8}
1023Biggs-Smith graph{3,2,2,2,1,1,1;1,1,1,1,1,1,3}
1265odd graph O_5{5,4,4,3;1,1,2,2}
208{12,10,5;1,1,8}
266{11,10,6,1;1,1,5,11}
280{9,8,6,4;1,1,3,8}
330{7,6,4,4;1,1,1,6}
4626odd graph O_6{6,5,5,4,4;1,1,2,2,3}
5{5,4,2;1,1,4}
{12,9,9;1,1,4}
{10,8,8,8;1,1,1,5}
{12,8,8,8;1,1,1,3}

See also

Automorphism Group, Distance-Transitive Graph

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References

Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, 1993.Biggs, N. L. "Automorphic Graphs and the Krein Condition." Geom. Dedicata 5, 117-127, 1976.Buekenhout, F. and Rowlinson, P. "The Uniqueness of Certain Automorphic Graphs." Geom. Dedicata 11, 443-446, 1981.Gordon, L. M. and Levingston, R. "The Construction of Some Automorphic Graphs." Geom. Dedicata 10, 261-267, 1981.Hall, J. I. "Locally Petersen Graphs." J. Graph Th. 4, 173-187, 1980.

Referenced on Wolfram|Alpha

Automorphic Graph

Cite this as:

Weisstein, Eric W. "Automorphic Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AutomorphicGraph.html

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