This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A363064 #23 Jun 27 2025 17:29:06 %S A363064 1,1,2,5,12,37,94,280,912,3164,8424 %N A363064 Number of connected Laplacian integral graphs on n vertices. %C A363064 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. %C A363064 Since every cograph is Laplacian integral, a(n) >= A000669(n). %H A363064 R. Grone and R. Merris, <a href="https://doi.org/10.1016/j.laa.2007.09.025">Indecomposable Laplacian integral graphs</a>, Linear Algebra and its Applications, 428 (2008), 1565-1570. %e A363064 For n <= 3, all connected graphs are Laplacian integral, so a(n) = A001349(n) when n <= 3. %e A363064 There is exactly one connected graph on 4 vertices that is not Laplacian integral: the path P_4, which has Laplacian matrix %e A363064 1 -1 0 0 %e A363064 -1 2 -1 0 %e A363064 0 -1 2 -1 %e A363064 0 0 -1 1 %e A363064 which has eigenvalues 0, 2, 2-sqrt(2), and 2+sqrt(2), which are not all integers. %Y A363064 Cf. A000669, A001349, A064731, A363065 (include disconnected graphs). %K A363064 nonn,hard,more %O A363064 1,3 %A A363064 _Nathaniel Johnston_, May 16 2023 %E A363064 a(10) from _M. A. Achterberg_, May 26 2023 %E A363064 a(11) from _Luis M. B. Varona_, Apr 27 2025