cp's OEIS Frontend

A184960 Irregular triangle C(n,g) read by rows, counting the connected 6-regular simple graphs on n vertices with girth exactly g.

%I A184960 #28 May 01 2014 02:37:01
%S A184960 1,1,4,21,266,7848,1,367860,0,21609299,1,1470293674,1,113314233799,9,
%T A184960 9799685588930,6
%N A184960 Irregular triangle C(n,g) read by rows, counting the connected 6-regular simple graphs on n vertices with girth exactly g.
%C A184960 The first column is for girth exactly 3. The row length sequence starts: 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3. The row length is incremented to g-2 when n reaches A054760(6,g).
%H A184960 Andries E. Brouwer, <a href="http://www.win.tue.nl/~aeb/graphs/cages/cages.html">Cages</a>
%H A184960 Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/C_k-reg_girth_eq_g_index">Index of sequences counting connected k-regular simple graphs with girth exactly g</a>
%H A184960 Jason Kimberley, <a href="/A184960/a184960_1.txt">Incomplete table of i, n, g, C(n,g)=a(i) for row n = 7..42</a>
%e A184960 Triangle begins:
%e A184960 1;
%e A184960 1;
%e A184960 4;
%e A184960 21;
%e A184960 266;
%e A184960 7848, 1;
%e A184960 367860, 0;
%e A184960 21609299, 1;
%e A184960 1470293674, 1;
%e A184960 113314233799, 9;
%e A184960 9799685588930, 6;
%e A184960 ?, 267;
%e A184960 ?, 3727;
%e A184960 ?, 483012;
%e A184960 ?, 69823723;
%e A184960 ?, 14836130862;
%e A184960 The C(40,5)=1 (see the a-file) graph, the unique (6,5)-cage, is the complement of a Petersen graph inside the Hoffman-Singleton graph [from Brouwer link].
%e A184960 The first known of C(42,5)>=1 graph(s) has automorphism group of order 5040 and these adjacency lists:
%e A184960 1 : 2 3 4 5 6 7
%e A184960 2 : 1 8 9 10 11 12
%e A184960 3 : 1 13 14 15 16 17
%e A184960 4 : 1 18 19 20 21 22
%e A184960 5 : 1 23 24 25 26 27
%e A184960 6 : 1 28 29 30 31 32
%e A184960 7 : 1 33 34 35 36 37
%e A184960 8 : 2 13 18 23 28 38
%e A184960 9 : 2 14 19 24 33 39
%e A184960 10 : 2 15 20 29 34 40
%e A184960 11 : 2 16 25 30 35 41
%e A184960 12 : 2 21 26 31 36 42
%e A184960 13 : 3 8 21 27 34 41
%e A184960 14 : 3 9 26 28 37 40
%e A184960 15 : 3 10 22 25 31 39
%e A184960 16 : 3 11 19 32 36 38
%e A184960 17 : 3 20 23 30 33 42
%e A184960 18 : 4 8 25 32 33 40
%e A184960 19 : 4 9 16 27 29 42
%e A184960 20 : 4 10 17 26 35 38
%e A184960 21 : 4 12 13 30 37 39
%e A184960 22 : 4 15 24 28 36 41
%e A184960 23 : 5 8 17 29 36 39
%e A184960 24 : 5 9 22 30 34 38
%e A184960 25 : 5 11 15 18 37 42
%e A184960 26 : 5 12 14 20 32 41
%e A184960 27 : 5 13 19 31 35 40
%e A184960 28 : 6 8 14 22 35 42
%e A184960 29 : 6 10 19 23 37 41
%e A184960 30 : 6 11 17 21 24 40
%e A184960 31 : 6 12 15 27 33 38
%e A184960 32 : 6 16 18 26 34 39
%e A184960 33 : 7 9 17 18 31 41
%e A184960 34 : 7 10 13 24 32 42
%e A184960 35 : 7 11 20 27 28 39
%e A184960 36 : 7 12 16 22 23 40
%e A184960 37 : 7 14 21 25 29 38
%e A184960 38 : 8 16 20 24 31 37
%e A184960 39 : 9 15 21 23 32 35
%e A184960 40 : 10 14 18 27 30 36
%e A184960 41 : 11 13 22 26 29 33
%e A184960 42 : 12 17 19 25 28 34
%Y A184960 Connected 6-regular simple graphs with girth at least g: A184961 (triangle); chosen g: A006822 (g=3), A058276 (g=4).
%Y A184960 Connected 6-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A184963 (g=3), A184964 (g=4).
%Y A184960 Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g: A198303 (k=3), A184940 (k=4), A184950 (k=5), this sequence (k=6), A184970 (k=7), A184980 (k=8).
%K A184960 nonn,hard,more,tabf
%O A184960 7,3
%A A184960 _Jason Kimberley_, Feb 24 2011
%E A184960 After approximately 390 processor days of computation with genreg, C(41,5)=0.