A363064 Number of connected Laplacian integral graphs on n vertices.

1, 1, 2, 5, 12, 37, 94, 280, 912, 3164

Offset

1,3

Comments

A (simple, undirected) graph is called Laplacian integral if all eigenvalues of its Laplacian matrix are integers. The corresponding sequence that uses the adjacency matrix instead of the Laplacian matrix is A064731.

Since every cograph is Laplacian integral, a(n) >= A000669(n).

Links

Table of n, a(n) for n=1..10.

R. Grone and R. Merris, Indecomposable Laplacian integral graphs, Linear Algebra and its Applications, 428 (2008), 1565-1570.

Example

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.

Crossrefs

Cf. A000669, A001349, A064731, A363065 (include disconnected graphs).

Sequence in context: A267398 A267399 A267400 * A052302 A280275 A009598

Adjacent sequences: A363061 A363062 A363063 * A363065 A363066 A363067

Keyword

nonn,hard,more

Author

Nathaniel Johnston, May 16 2023

Extensions

a(10) from M. A. Achterberg, May 26 2023

Status

approved