A076184 -id:A076184 - cp's OEIS frontend

A382194 List of connected graphs that are squares, encoded as in A076184.

0, 1, 7, 31, 63, 239, 255, 511, 1023, 3455, 3887, 3951, 3967, 4095, 7679, 7903, 7935, 8191, 16350, 16351, 16383, 32767, 104063, 104447, 106287, 106351, 111587, 111599, 112511, 112623, 112639, 127791, 127855, 127871, 128879, 128895, 129023, 131071, 237567

Offset: 1

Pontus von Brömssen, Mar 18 2025

Intersection of A382193 and A382195.

			As an irregular triangle, where row n >= 1 contains A382180(n) terms:
     0;
     1;
     7;
    31,   63;
   239,  255,  511, 1023;
  3455, 3887, 3951, 3967, 4095, 7679, 7903, 7935, 8191, 16350, 16351, 16383, 32767;
  ...
The diamond graph is connected and isomorphic to the square of the path graph on 4 vertices. The code of the diamond graph is 31, so 31 is a term.

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

Cf. A076184, A382180, A382193, A382195, A382283.

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.

A382192 Number of components of the graph with code A076184(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

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:
  1;
  1;
  1, 1;
  1, 2, 1, 1, 1, 1, 1;
  1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  ...
For n = 6, A076184(6) = 12 encodes the graph on 4 vertices and 2 disjoint edges. This graph has 2 components, so a(6) = 2.

Cf. A002494, A076184, A382193.

A382193 List of connected graphs, encoded as in A076184.

Original entry on oeis.org

0, 1, 3, 7, 11, 13, 15, 30, 31, 63, 75, 77, 79, 86, 87, 94, 95, 117, 119, 127, 222, 223, 235, 236, 237, 239, 254, 255, 507, 511, 1023, 1099, 1101, 1103, 1109, 1110, 1111, 1118, 1119, 1141, 1143, 1151, 1182, 1183, 1187, 1191, 1195, 1196, 1197, 1198, 1199, 1214

Offset: 1

Pontus von Brömssen, Mar 18 2025

Numbers A076184(k) such that A382192(k) = 1.

			As an irregular triangle, where row n >= 1 contains A001349(n) terms:
   0;
   1;
   3,  7;
  11, 13, 15, 30, 31, 63;
  75, 77, 79, 86, 87, 94, 95, 117, 119, 127, 222, 223, 235, 236, 237, 239, 254, 255, 507, 511, 1023;
  ...

Cf. A001349, A076184, A382192.

A382756 a(n) is the graph corresponding to A076184(n), encoded as in A382754.

Original entry on oeis.org

1, 3, 11, 15, 71, 76, 77, 79, 94, 95, 127, 1039, 1052, 1053, 1055, 1082, 1083, 1086, 1087, 1208, 1209, 1211, 1215, 1150, 1151, 1231, 1244, 1245, 1247, 1278, 1279, 1519, 1535, 2047, 32799, 32828, 32829, 32831, 32888, 32889, 32890, 32891, 32894, 32895, 33400

Offset: 1

Pontus von Brömssen, Apr 04 2025

			As an irregular triangle, where the first row contains 1 term and row n >= 2 contains A002494(n) terms:
   1;
   3;
  11, 15;
  71, 76, 77, 79, 94, 95, 127;
  ...

Cf. A002494, A076184, A382754, A382757.

A382757 a(n) is the graph corresponding to A382754(n), encoded as in A076184.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 7, 0, 1, 3, 11, 7, 12, 13, 15, 30, 31, 63, 0, 1, 3, 11, 75, 7, 12, 13, 15, 76, 77, 79, 30, 31, 86, 87, 94, 95, 222, 223, 63, 116, 117, 119, 127, 235, 236, 237, 239, 254, 255, 507, 511, 1023, 0, 1, 3, 11, 75, 1099, 7, 12, 13, 15, 76, 77, 79

Offset: 1

Pontus von Brömssen, Apr 04 2025

Any isolated vertices in the graphs are ignored (except for the 1-vertex graph).

			As an irregular triangle, where row n >= 1 contains A000088(n) terms:
  0;
  0, 1;
  0, 1, 3,  7;
  0, 1, 3, 11, 7, 12, 13, 15, 30, 31, 63;
  ...

Cf. A000088, A076184, A382754, A382756.

A382191 Number of edges of the graph with code A076184(n).

Original entry on oeis.org

0, 1, 2, 3, 3, 2, 3, 4, 4, 5, 6, 4, 3, 4, 5, 4, 5, 5, 6, 4, 5, 6, 7, 6, 7, 6, 5, 6, 7, 7, 8, 8, 9, 10, 5, 4, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 7, 8, 6, 7, 3, 4, 5, 6, 5, 6, 5, 6, 6, 7, 7, 8, 7, 8, 7, 6, 7, 8, 6, 7, 7, 8, 8, 9, 5, 6, 7, 6, 7, 8, 7, 8, 7, 8, 9, 9, 8

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;
  2, 3;
  3, 2, 3, 4, 4, 5, 6;
  4, 3, 4, 5, 4, 5, 5, 6, 4, 5, 6, 7, 6, 7, 6, 5, 6, 7, 7, 8, 8, 9, 10;
  ...

Cf. A000120, A002494, A076184.

a(n) = A000120(A076184(n)).

A382281 Let n encode the edges of a graph by taking edges (u,v), with u < v, in colexicographic order ((0,1), (0,2), (1,2), (0,3), ...) and adding each edge to the graph if the corresponding binary digit of n (starting with the least significant digit) is 1. a(n) is the smallest nonnegative integer that encodes the same unlabeled graph as n (disregarding any isolated vertices), i.e., the code of the graph as defined in A076184.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 3, 7, 1, 3, 3, 11, 12, 13, 13, 15, 1, 3, 12, 13, 3, 11, 13, 15, 3, 7, 13, 15, 13, 15, 30, 31, 1, 12, 3, 13, 3, 13, 11, 15, 3, 13, 7, 15, 13, 30, 15, 31, 3, 13, 13, 30, 7, 15, 15, 31, 11, 15, 15, 31, 15, 31, 31, 63, 1, 3, 3, 11, 12, 13, 13, 15

Offset: 0

Pontus von Brömssen, Mar 21 2025

nonn

			n = 6 is 110 in binary, encoding the graph with edges (0,2) and (1,2), i.e., the path graph on 3 vertices. The canonical code of that graph is a(6) = 3, corresponding to the graph with edges (0,1) and (0,2).

Cf. A076184.

a(n) <= n with equality if and only if n is in A076184.

A382282 Code for the n-dimensional hypercube graph, encoded as in A076184.

Original entry on oeis.org

0, 1, 30, 15054720, 608598583690983931143264520896512

Offset: 0

Pontus von Brömssen, Mar 21 2025

Eric Weisstein's World of Mathematics, Hypercube Graph.
Wikipedia, Hypercube graph.

Cf. A076184.

A382754 List of unlabeled simple graphs, encoded as integers (see comments).

Original entry on oeis.org

0, 1, 2, 3, 8, 9, 11, 15, 64, 65, 67, 71, 75, 76, 77, 79, 94, 95, 127, 1024, 1025, 1027, 1031, 1039, 1043, 1044, 1045, 1047, 1052, 1053, 1055, 1078, 1079, 1082, 1083, 1086, 1087, 1150, 1151, 1207, 1208, 1209, 1211, 1215, 1231, 1244, 1245, 1247, 1278, 1279, 1519, 1535, 2047

Offset: 0

Pontus von Brömssen, Apr 04 2025

For a graph G, pick a permutation of its vertices that minimizes the bitstring obtained by reading the lower triangular part of the corresponding adjacency matrix by rows. The code of G is that bitstring interpreted as a binary number plus 2^(v*(v-1)/2), where v is the number of vertices of G; see example. As a special case, the code of the null graph is 0. The sequence consists of all such minimal codes.
For n >= 1, the numbers of vertices and edges of the graph with code a(n) are A002024(A000523(a(n))+1) and A000120(a(n))-1 = A382758(n), respectively.
This sequence can be used to define sequences for:
  - graph invariants (examples: A382758, A382759, A382760);
  - graph operators, either by code (A382763) or by index (A382764);
  - lists of subsets of graphs, either by code (A382761) or by index (A382762).

			As an irregular triangle, where row n >= 0 contains A000088(n) terms:
   0;
   1;
   2,  3;
   8,  9, 11, 15;
  64, 65, 67, 71, 75, 76, 77, 79, 94, 95, 127;
  ...
71 is a term, because it is the code of the claw graph. If the edges are taken to be (0,1), (0,2), and (0,3), an optimal permutation of the vertices of the graph is (3, 2, 1, 0), with the lower triangular part of the corresponding adjacency matrix being [0; 0,0; 1,1,1]. Adding 2^(4*3/2) to the binary number 000111, we obtain that the code of the claw graph is 64+7 = 71.

Pontus von Brömssen, Table of n, a(n) for n = 0..13598 (for graphs on up to 8 vertices)

Cf. A000088, A000120, A000523, A002024, A076184, A382755, A382756, A382757, A382758, A382759, A382760, A382761, A382762, A382763, A382764.

cp's OEIS Frontend

A382194 List of connected graphs that are squares, encoded as in A076184.

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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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A382192 Number of components of the graph with code A076184(n).

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A382193 List of connected graphs, encoded as in A076184.

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A382756 a(n) is the graph corresponding to A076184(n), encoded as in A382754.

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A382757 a(n) is the graph corresponding to A382754(n), encoded as in A076184.

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A382191 Number of edges of the graph with code A076184(n).

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A382282 Code for the n-dimensional hypercube graph, encoded as in A076184.

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A382754 List of unlabeled simple graphs, encoded as integers (see comments).

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