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A263341 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs on n vertices with independence number k.

%I A263341 #64 Feb 16 2025 08:33:27
%S A263341 1,1,1,1,2,1,1,6,3,1,1,13,15,4,1,1,37,82,30,5,1,1,106,578,301,51,6,1,
%T A263341 1,409,6021,4985,842,80,7,1,1,1896,101267,142276,27107,1995,117,8,1,1,
%U A263341 12171,2882460,7269487,1724440,112225,4210,164,9,1,1,105070,138787233,655015612,210799447,13893557,388547,8165,221,10,1
%N A263341 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs on n vertices with independence number k.
%C A263341 The independence number of a graph is the maximum size of an independent set.
%C A263341 Row sums give A000088, n >= 1.
%C A263341 T(n,k) is also the number of graphs on n vertices such that a largest clique is of size k. - _Geoffrey Critzer_, Sep 23 2016
%C A263341 T(n,k) is also the number of graphs on n vertices such that the size of a smallest vertex cover is n-k. - _Geoffrey Critzer_, Sep 23 2016
%C A263341 T(n,k) is also the number of graphs on n vertices with independence number k. - _Eric W. Weisstein_, May 17 2017
%C A263341 For any graph the independence number is greater than or equal to the independent domination number (A332402) and less than or equal to the upper domination number (A332403). - _Andrew Howroyd_, Feb 19 2020
%H A263341 Brendan McKay, <a href="/A263341/b263341.txt">Table of n, a(n) for n = 1..91</a> (first 13 rows)
%H A263341 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000093">The length of the maximal independent set of vertices of a graph</a>.
%H A263341 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000097">The order of the largest clique of the graph</a>.
%H A263341 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CliqueNumber.html">Clique Number</a>
%H A263341 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependenceNumber.html">Independence Number</a>
%H A263341 Wikipedia, <a href="https://en.wikipedia.org/wiki/Clique_%28graph_theory%29">Clique (graph theory)</a>
%e A263341 Triangle begins:
%e A263341   1;
%e A263341   1,     1;
%e A263341   1,     2,       1;
%e A263341   1,     6,       3,       1;
%e A263341   1,    13,      15,       4,       1;
%e A263341   1,    37,      82,      30,       5,      1;
%e A263341   1,   106,     578,     301,      51,      6,    1;
%e A263341   1,   409,    6021,    4985,     842,     80,    7,   1;
%e A263341   1,  1896,  101267,  142276,   27107,   1995,  117,   8, 1;
%e A263341   1, 12171, 2882460, 7269487, 1724440, 112225, 4210, 164, 9, 1;
%e A263341   ...
%Y A263341 Row sums are A000088.
%Y A263341 Columns 2..9 are A052450, A052451, A052452, A077392, A077393, A077394, A205577, A205578.
%Y A263341 Transpose of A287024.
%Y A263341 Cf. A115196, A126744 (clique number of connected graphs), A294490 (independence number of connected graphs).
%Y A263341 Cf. A263284, A332402, A332403.
%K A263341 nonn,tabl
%O A263341 1,5
%A A263341 _Christian Stump_, Oct 15 2015
%E A263341 a(21)-a(28) from _Geoffrey Critzer_, Sep 22 2016
%E A263341 Rows 8-10 from _Eric W. Weisstein_, May 16 2017
%E A263341 Rows 11-13 from _Brendan McKay_, Feb 18 2020
%E A263341 Name clarified by _Andrew Howroyd_, Feb 18 2020