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 A076263 Triangle read by rows: T(n,k) = number of nonisomorphic connected graphs with n vertices and k edges (n >= 1, n-1 <= k <= n(n-1)/2). 3
 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 5, 4, 2, 1, 1, 6, 13, 19, 22, 20, 14, 9, 5, 2, 1, 1, 11, 33, 67, 107, 132, 138, 126, 95, 64, 40, 21, 10, 5, 2, 1, 1, 23, 89, 236, 486, 814, 1169, 1454, 1579, 1515, 1290, 970, 658, 400, 220, 114, 56, 24, 11, 5, 2, 1, 1, 47, 240, 797, 2075, 4495 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS The index of the T(n,k) in the sequence is ((n-2)^3 - n + 6*k + 8)/6. T(n,k)=1 for k = n*(n-1)/2-1 and k = n*(n-1)/2 (therefore {1,1} separates sublists for given numbers of vertices (n > 2)). LINKS T. D. Noe, Rows 1 to 16 of triangle, flattened (from Gordon Royle's website) Keith M. Briggs, Combinatorial Graph Theory. Sriram V. Pemmaraju, The Combinatorica Project Marko R. Riedel, Number of distinct connected digraphs, Math StackExchange. Eric Weisstein's World of Mathematics, Connected Graph. EXAMPLE 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 = 5. Triangle begins:    1;    1;    1,  1;    2,  2,  1,   1;    3,  5,  5,   4,   2,   1,   1;    6, 13, 19,  22,  20,  14,   9,  5,  2,  1,  1;   11, 33, 67, 107, 132, 138, 126, 95, 64, 40, 21, 10, 5, 2, 1, 1; MATHEMATICA NumberOfConnectedGraphs[vertices_, edges_] := Plus @@ ConnectedQ /@ ListGraphs[vertices, edges] /. {True->1, False ->0} (* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) Table[Plus @@ ConnectedQ /@ ListGraphs[Vert, i] /. {True -> 1, False -> 0}, {Vert, 8}, {i, Vert - 1, Vert*(Vert - 1)/2}] CROSSREFS Row lengths (excluding first row): A000124. Number of connected graphs for given number of vertices: A001349. Number of connected graphs for given number of edges: A002905. Number of entries in the n-th row is A152947. Row sums give A001349. Starting each row from k=0 gives A054924, which is the main entry for this triangle. Sequence in context: A344567 A076037 A215563 * A272689 A274887 A008302 Adjacent sequences:  A076260 A076261 A076262 * A076264 A076265 A076266 KEYWORD nonn,tabf AUTHOR Arne Ring (arne.ring(AT)epost.de), Oct 03 2002 EXTENSIONS Corrected by Keith Briggs and Robert G. Wilson v, May 01 2005 Rows 5, 6 & 7 from Robert G. Wilson v, Jun 21 2005 More terms from Keith Briggs, Jun 28 2005 Name corrected by Andrey Zabolotskiy, Nov 20 2017 STATUS approved

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Last modified January 28 18:02 EST 2022. Contains 350657 sequences. (Running on oeis4.)