%I A008406
%S A008406 1,1,1,1,1,1,1,1,1,2,3,2,1,1,1,1,2,4,6,6,6,4,2,1,1,1,1,2,5,9,
%T A008406 15,21,24,24,21,15,9,5,2,1,1,1,1,2,5,10,21,41,65,97,131,148,
%U A008406 148,131,97,65,41,21,10,5,2,1,1,1,1,2,5,11,24,56,115,221
%N A008406 Triangle T(n,k) read by rows, giving number of graphs with n nodes (n 
               >= 1) and k edges (0 <= k <= n(n-1)/2).
%C A008406 The index of T(n,k) in the sequence is ((n-2)^3-n+6*k+8)/6).
%C A008406 T(n,k)=1 for k=n*(n-1)/2-1 and k=n*(n-1)/2 (therefore the pair {1,1} 
               separates sublists for a given number of vertices (n>2).
%D A008406 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 264.
%D A008406 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 
               2004; p. 519.
%D A008406 F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 214.
%D A008406 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 
               1973, p. 240.
%D A008406 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 
               146.
%D A008406 R. W. Robinson, Numerical implementation of graph counting algorithms, 
               AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
%H A008406 R. W. Robinson, <a href="b008406.txt">Rows 1 to 20 of triangle, flattened</
               a>
%H A008406 Sriram V. Pemmaraju, <a href="http://www.cs.uiowa.edu/~sriram/Combinatorica/
               index.html">Combinatorica 2.0</a>
%H A008406 Gordon Royle, <a href="http://www.cs.uwa.edu.au/~gordon/remote/graphs/
               index.html#nums">Small graphs</a>
%H A008406 S. S. Skiena, <a href="http://www.cs.sunysb.edu/~algorith/files/generating-graphs.shtml">
               Generating graphs</a>
%H A008406 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SimpleGraph.html">Simple Graph</a>
%H A008406 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ConnectedGraph.html">Link to a section of The World of Mathematics.</
               a>
%e A008406 There are 2 connected graphs with 4 vertices and 3 edges up to isomorphy 
               (first graph: ((1,2),(2,3),(3,4)); second graph: (1,2),(1,3),(1,4)) 
               ). Index within the sequence is ((4-2)^3-4 +6*3+8)/6))=6.
%e A008406 Triangle begins:
%e A008406 1,
%e A008406 1,1,
%e A008406 1,1,1,1,
%e A008406 1,1,2,3,2,1,1, [graphs with 4 nodes and from 0 to 6 edges]
%e A008406 1,1,2,4,6,6,6,4,2,1,1,
%e A008406 1,1,2,5,9,15,21,24,24,21,15,9,5,2,1,1,
%e A008406 1,1,2,5,10,21,41,65,97,131,148,148,131,97,65,41,21,10,5,2,1,1,
%e A008406 ...
%t A008406 NumberOfConnectedGraphs[vertices_, edges_] := Apply[Plus, Map[ConnectedQ, 
               ListGraphs[vertices, edges]]/.{True->1}/.{False ->0}] The sequence 
               up to MAX vertices: Table[Table[Apply[Plus, Map[ConnectedQ, ListGraphs[Vert, 
               i]]] /.{True->1}/.{False->0}, {i, Vert-1, Vert*(Vert-1)/2}], {Vert, 
               1, MAX}]] Requires Combinatorica 2.0
%Y A008406 Row sums give A000088.
%Y A008406 Cf. A046742, A046751, A000664, A000717, A001432, A001431, A001430, A001433, 
               A001434, A048179, A048180 etc.
%Y A008406 Cf. also A039735.
%Y A008406 Index calculation for the current sequence: A000124, number of connected 
               graphs for given number of vertices: A001349, number of connected 
               graphs for given number of edges: A076265, number of disconnected 
               graphs for given number of vertices and edges: A076264.
%Y A008406 Sequence in context: A110540 A083475 A122402 this_sequence A039735 A129385 
               A096626
%Y A008406 Adjacent sequences: A008403 A008404 A008405 this_sequence A008407 A008408 
               A008409
%K A008406 nonn,tabf,nice
%O A008406 1,10
%A A008406 N. J. A. Sloane (njas(AT)research.att.com).
%E A008406 Additional comments from Arne Ring (arne.ring(AT)epost.de), Oct 03 2002