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%I #20 Mar 13 2022 05:48:04
%S 1,1,1,2,2,4,2,5,4,8,2,11,4,8,10
%N Number of distinct connected simple symmetric (edge- and vertex-transitive) graphs with n nodes.
%C Care is needed with "symmetric" terminology, which is variously used to mean both arc-transitive and both vertex- and edge-transitive.
%C The symmetry means that any two vertices and any two edges are equivalent. In other words, if we have an initial labeling of the graph with vertices A and B adjacent (directly connected by an edge), we can relabel any two adjacent vertices as A and B and then relabel the remaining vertices so that new graph will be equal to the initial.
%C The first known difference from A286280 (connected arc-transitive graphs on n vertices) occurs at a(27), corresponding to the Doyle graph (which is both edge- and vertex-transitive but not arc-transitive). - _Eric W. Weisstein_, May 13 2017
%C By convention, empty graphs are considered edge-transitive (and hence symmetric).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Arc-TransitiveGraph.html">Arc-Transitive Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DoyleGraph.html">Doyle Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Edge-TransitiveGraph.html">Edge-Transitive Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SymmetricGraph.html">Symmetric Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Vertex-TransitiveGraph.html">Vertex-Transitive Graph</a>
%e The complete graph is symmetrical.
%e In addition, if the number of vertices is > 3, the simple cycle through all vertices is symmetrical.
%e Graphs determined by vertices and edges of Platonic solids are symmetrical.
%e The square K X K grid with right vertices connected to corresponding left vertices and bottom vertices connected to corresponding top vertices is symmetrical.
%e The smallest nontrivial and non-Platonic symmetric graph is the hexagon with connected opposite vertices.
%e An example of a symmetrical graph with 13 vertices:
%e 0 connected to 1, 2, 3, 4
%e 1 connected to 0, 5, 6, 7
%e 2 connected to 0, 5, 8, 9
%e 3 connected to 0, 6, 10, 11
%e 4 connected to 0, 8, 10, 12
%e 5 connected to 1, 2, 10, 11
%e 6 connected to 1, 3, 8, 12
%e 7 connected to 1, 8, 9, 11
%e 8 connected to 2, 4, 6, 7
%e 9 connected to 2, 7, 10, 12
%e 10 connected to 3, 4, 5, 9
%e 11 connected to 3, 5, 7, 12
%e 12 connected to 4, 6, 9, 11
%Y Cf. A087145 (number of disconnected simple symmetric graphs on n nodes).
%Y Cf. A286931 (number of not necessarily connected simple symmetrical graphs with n nodes).
%Y Cf. A286280 (number of connected arc-transitive graphs of order n).
%K hard,more,nice,nonn
%O 1,4
%A Eugene Vasilchenko (eugene(AT)vasilchenko.net), Oct 10 2007, Oct 14 2007
%E a(1) and a(2) changed from 0 to 1 (since K_1 and K_2 are connected, vertex-transitive, and edge-transitive) by _Eric W. Weisstein_, May 16 2017
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