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a(n) is the number of full graphs on n unlabeled vertices (see comments for definition of full graph).
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%I #42 Oct 08 2023 09:17:49

%S 1,1,2,3,10,32,154,1037,12339,274640

%N a(n) is the number of full graphs on n unlabeled vertices (see comments for definition of full graph).

%C Full graphs are defined as follows: A tree T with n vertices is bipartite, let its parts be of size k and n - k with k <= n-k, we rename T as T_k and we say it has type (k, n - k). If a graph G has a spanning tree T_k of type (k,n-k) for each k such that 1 <= k <= [n/2], then we say that G is full.

%C Full graphs will be featured in an upcoming article by Jayasooriya, McSorley, and Schuerger.

%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a364/A364208.java">Java program</a> (github)

%e a(4)=3 as follows. Of the eleven nonisomorphic graphs on four vertices, only four of them contain a subgraph isomorphic to K_{1,3}, and thus a spanning tree of type (1,3). One of these four graphs does not contain a spanning tree of type (2,2), while the other three graphs each contain a Hamiltonian path and thus a spanning tree of type (2,2).

%K nonn,more

%O 1,3

%A _Houston Schuerger_, Aug 26 2023

%E a(8)-a(10) from _Sean A. Irvine_, Oct 07 2023