cp's OEIS Frontend

A301425 Number of plane 5-regular simple connected graphs with 2n vertices.

%I A301425 #30 Feb 16 2025 08:33:53
%S A301425 1,0,1,1,6,14,98,529,4035,31009,252386,2073769,17277113
%N A301425 Number of plane 5-regular simple connected graphs with 2n vertices.
%C A301425 We count here plane graphs, i.e., graphs embedded in the plane, up to embedding-preserving isomorphism, while such sequences as A003094 count planar graphs (counted up to abstract isomorphism). In this we follow the nomenclature of Brendan McKay, cf. link.
%H A301425 B. D. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/data/planegraphs.html">Plane graphs</a> (see also <a href="http://users.cecs.anu.edu.au/~bdm/data/graphs.html">the section on Planar Graphs on this page</a>).
%H A301425 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IcosahedralGraph.html">Icosahedral Graph</a>.
%H A301425 <a href="/index/Gra#graphs">Index to sequences related to planar graphs</a>
%e A301425 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}.
%e A301425 For other numbers of vertices, the number of plane 5-regular simple connected graphs is as follows:
%e A301425 14 vertices: 0  graphs,
%e A301425 16 vertices: 1  graph,
%e A301425 18 vertices: 1  graph,
%e A301425 20 vertices: 6  graphs,
%e A301425 22 vertices: 14  graphs,
%e A301425 24 vertices: 98  graphs,
%e A301425 26 vertices: 529  graphs,
%e A301425 28 vertices: 4035  graphs,
%e A301425 30 vertices: 31009  graphs,
%e A301425 32 vertices: 252386  graphs,
%e A301425 34 vertices: 2073769 graphs,
%e A301425 36 vertices: 17277113 graphs. (From the McKay web page.)
%Y A301425 Cf. A308489, A292109, A292972, A323281, A322917, A322929, A006821, A003094.
%K A301425 nonn,more
%O A301425 6,5
%A A301425 _M. F. Hasler_, Mar 20 2018