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EXAMPLE
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There is only a(6) = 1 planar 5-regular simple connected graph with 2n = 12 vertices, which is the icosahedral graph, cf. MathWorld link. If we label the vertices 1, ..., 9, A, B, C, they are connected as follows: 1 -> {2 3 4 5 6}, 2 -> {1 6 7 8 3}, 3 -> {1 2 8 9 4}, 4 -> {1 3 9 A 5}, 5 -> {1 4 A B 6}, 6 -> {1 5 B 7 2 }, 7 -> {2 6 B C 8}, 8 -> {2 7 C 9 3}, 9 -> {3 8 C A 4}, A -> {4 9 C B 5}, B -> {5 A C 7 6}, C -> {7 B A 9 8}.
For other numbers of vertices, the number of plane 5-regular simple connected graphs is as follows:
14 vertices: 0 graphs,
16 vertices: 1 graph,
18 vertices: 1 graph,
20 vertices: 6 graphs,
22 vertices: 14 graphs,
24 vertices: 98 graphs,
26 vertices: 529 graphs,
28 vertices: 4035 graphs,
30 vertices: 31009 graphs,
32 vertices: 252386 graphs,
34 vertices: 2073769 graphs,
36 vertices: 17277113 graphs. (From the McKay web page.)
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