A382194 -id:A382194 - cp's OEIS frontend

A382283 Number of square roots of connected square graphs in the order listed in A382194.

1, 1, 2, 1, 5, 1, 2, 3, 15, 1, 1, 2, 3, 4, 1, 3, 3, 15, 1, 1, 17, 60, 1, 2, 1, 2, 1, 1, 1, 1, 4, 2, 3, 2, 4, 11, 10, 11, 2, 1, 5, 3, 3, 6, 9, 8, 6, 1, 1, 19, 51, 3, 21, 1, 1, 3, 21, 2, 3, 113, 1, 11, 127, 374, 1, 1, 2, 3, 4, 1, 1, 2, 3, 4, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Pontus von Brömssen, Mar 22 2025

A382194 lists all connected graphs that are squares, encoded as in A076184. a(n) is the number of unlabeled graphs whose squares are isomorphic to the n-th graph in A382194.

			As an irregular triangle, where row n >= 1 contains A382180(n) terms:
  1;
  1;
  2;
  1, 5;
  1, 2, 3, 15;
  1, 1, 2,  3, 4, 1, 3, 3, 15, 1, 1, 17, 60;
  ...
The last term on row n equals A241706(n)+1, the number of graphs whose square is the complete graph on n vertices.

Pontus von Brömssen, Table of n, a(n) for n = 1..1580 (for graphs on up to 9 vertices)
Eric Weisstein's World of Mathematics, Graph Square.
Wikipedia, Graph power.

Cf. A076184, A241706, A382180, A382194.

A382180 Number of unlabeled connected graphs with n vertices which are squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 13, 42, 206, 1310, 12622, 180700, 3925282

Offset: 0

Brendan McKay and Sean A. Irvine, Mar 17 2025

If G is an unlabeled finite simple graph, define its square S(G) to be the graph with the same vertices as G. The edges of S(G) are the edges of G together with an edge from vertex u to v whenever u and v are not adjacent in G but are joined by a path of length 2. [There is an obvious generalization to the square of a directed graph.- N. J. A. Sloane, Mar 24 2025]
The present definition, the number of unlabeled connected graphs with n vertices which are squares, implies "which are squares of connected graphs on n vertices", since if G is not connected, neither is its square. - N. J. A. Sloane, Mar 24 2025.
If the squares of two trees are isomorphic, then the trees themselves are isomorphic [Ross and Harary]. Thus the number of squares of trees is the same as the number of trees, A000055.

Frank Harary and Ian C. Ross, The Square of a Tree, Bell Labs Memorandum MM-59-122-2, May 16, 1959, 11 pages.

FindStat - The combinatorial statistics database, The square of a graph.
A. Mukhopadhyay, The Square Root of a Graph, J Comb. Th., 2, (1967), 290-295.
Ian C. Ross and Frank Harary, The square of a tree, The Bell System Technical J, 39, 3, May 1960.
Eric Weisstein's World of Mathematics, Graph Square.
Wikipedia, Graph power.

Cf. A000055, A001349, A382194.

Inverse Euler transform of A382181.

A382195 a(n) is the code (in the encoding given by A076184) of the square of the graph with code A076184(n).

Original entry on oeis.org

0, 1, 7, 7, 63, 12, 31, 63, 63, 63, 63, 1023, 116, 255, 1023, 239, 511, 511, 1023, 116, 255, 511, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 32767, 1972, 4095, 32767, 3873, 7903, 3951, 8191, 8191, 32767, 3873, 7903, 8191, 32767

Offset: 1

Pontus von Brömssen, Mar 18 2025

			As an irregular triangle, where the first row contains 1 term and row n >= 2 contains A002494(n) terms:
  0;
  1;
  7, 7;
  63, 12, 31, 63, 63, 63, 63;
  ...
For n = 7, A076184(7) = 13 is the code for the path graph on 4 vertices. The square of that graph is the diamond graph, whose code is 31 = a(7).

FindStat - The combinatorial statistics database, The square of a graph.
Eric Weisstein's World of Mathematics, Graph Square.
Wikipedia, Graph power.

Cf. A002494, A076184, A382194.

A382761 List of graphs that are squares, encoded as in A382754.

Original entry on oeis.org

0, 1, 2, 3, 8, 9, 15, 64, 65, 75, 76, 95, 127, 1024, 1025, 1043, 1044, 1079, 1207, 1208, 1247, 1279, 1535, 2047, 32768, 32769, 32803, 32804, 32871, 33383, 33384, 33424, 33455, 33519, 33689, 34543, 34687, 36863, 38639, 38640, 38673, 38711, 38719, 38783, 38911, 39423, 39935, 40959, 48638, 48639, 49151, 65535

Offset: 0

Pontus von Brömssen, Apr 05 2025

			As an irregular triangle, where row n >= 0 contains A382181(n) terms:
     0;
     1;
     2,    3;
     8,    9,   15;
    64,   65,   75,   76,   95,  127;
  1024, 1025, 1043, 1044, 1079, 1207, 1208, 1247, 1279, 1535, 2047;
  ...

Cf. A382181, A382194, A382754, A382762, A382763.

a(n) = A382754(A382762(n)).

A382762 List of graphs that are squares, encoded by their indices in A382754.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 17, 18, 19, 20, 24, 25, 32, 39, 40, 47, 49, 51, 52, 53, 54, 59, 60, 70, 89, 90, 99, 107, 117, 127, 144, 160, 170, 171, 172, 177, 182, 186, 191, 193, 195, 201, 204, 205, 206, 207, 208

Offset: 0

Pontus von Brömssen, Apr 05 2025

			As an irregular triangle, where row n >= 0 contains A382181(n) terms:
   0;
   1;
   2,  3;
   4,  5,  7;
   8,  9, 12, 13, 17, 18;
  19, 20, 24, 25, 32, 39, 40, 47, 49, 51, 52;
  ...

Cf. A382181, A382194, A382754, A382761, A382764.

cp's OEIS Frontend

A382283 Number of square roots of connected square graphs in the order listed in A382194.

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A382180 Number of unlabeled connected graphs with n vertices which are squares.

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A382195 a(n) is the code (in the encoding given by A076184) of the square of the graph with code A076184(n).

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A382761 List of graphs that are squares, encoded as in A382754.

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A382762 List of graphs that are squares, encoded by their indices in A382754.

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