A241842
Number of simple connected graphs on n nodes that are non-integral.
Original entry on oeis.org
0, 0, 1, 4, 18, 106, 846, 11095, 261056, 11716488, 1006700452
Offset: 1
A077027
Number of simple integral graphs on n nodes.
Original entry on oeis.org
1, 2, 3, 6, 10, 20, 33, 71, 121, 269, 492
Offset: 1
Cf.
A064731 (number of simple connected integral graphs).
Cf.
A287154 (number of simple disconnected integral graphs).
Cf.
A363065 (number of Laplacian integral graphs).
A363064
Number of connected Laplacian integral graphs on n vertices.
Original entry on oeis.org
1, 1, 2, 5, 12, 37, 94, 280, 912, 3164, 8424
Offset: 1
For n <= 3, all connected graphs are Laplacian integral, so a(n) = A001349(n) when n <= 3.
There is exactly one connected graph on 4 vertices that is not Laplacian integral: the path P_4, which has Laplacian matrix
1 -1 0 0
-1 2 -1 0
0 -1 2 -1
0 0 -1 1
which has eigenvalues 0, 2, 2-sqrt(2), and 2+sqrt(2), which are not all integers.
A287154
Number of simple disconnected integral graphs on n vertices.
Original entry on oeis.org
0, 1, 2, 4, 7, 14, 26, 49, 97, 186, 379
Offset: 1
Cf.
A077027 (number of simple not necessarily connected integral graphs).
Cf.
A064731 (number of simple connected integral graphs).
A242952
Number of connected graphs on n vertices whose spectrum has n distinct eigenvalues.
Original entry on oeis.org
1, 1, 1, 3, 11, 54, 539, 7319, 209471, 10000304
Offset: 1
- Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
- Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
- Haiying Shan and Xiaoqi Liu, Exploring Graphs with Distinct M-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability, arXiv:2412.18759 [math.CO], 2024. See pp. 14-15.
- Eric Weisstein's World of Mathematics, Graph Spectrum
Cf.
A064731 (integral graphs),
A242953 (non-distinct spectrum graphs).
Corrected, original description as the "real spectrum" was incorrect, by
Travis Hoppe, Mar 23 2015
A242953
Number of connected graphs on n vertices whose spectrum has fewer than n distinct eigenvalues.
Original entry on oeis.org
0, 0, 1, 3, 10, 58, 314, 3798, 51609, 1716267
Offset: 1
Corrected, original description as the "non-real spectrum" was incorrect, by
Travis Hoppe, Mar 23 2015
A243332
Number of simple connected graphs with n nodes that are integral and triangle-free.
Original entry on oeis.org
1, 1, 0, 1, 1, 3, 1, 3, 0, 14, 8, 18, 33, 75
Offset: 1
A243786
Number of unlabeled simple connected graphs with n nodes that are chordal and integral.
Original entry on oeis.org
1, 1, 1, 1, 3, 2, 5, 9, 2, 14
Offset: 1
A243273
Number of unlabeled simple graphs with n nodes that are Hamiltonian and integral.
Original entry on oeis.org
0, 0, 0, 1, 7, 43, 379, 6185, 177071, 9305068
Offset: 1
A243274
Number of graphs with n nodes that are Hamiltonian and non-integral.
Original entry on oeis.org
1, 0, 1, 2, 1, 5, 4, 11, 12, 50
Offset: 1
Showing 1-10 of 19 results.
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