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%I #7 Apr 26 2021 21:32:31
%S 1,0,1,0,0,2,0,0,2,4,0,0,0,13,8,0,0,0,13,83,16,0,0,0,0,308,513,32,0,0,
%T 0,0,296,7460,3297,64,0,0,0,0,68,38630,200207,22047,128,0,0,0,0,0,
%U 65211,4788564,6709094,153446,256
%N Triangle read by rows, 1 <= k <= n: T(n,k) is the number of (unlabeled) connected graphs with n nodes and throttling number k.
%C The throttling number of a graph can be defined as follows. Start with a blue/white coloring of the nodes. At each step, all white nodes, which are currently the unique white neighbor of a blue node, are colored blue. The throttling number is the minimum, over all possible initial colorings, of the sum of the number of blue nodes in the initial coloring and the number of steps required to color all nodes blue.
%C A connected graph with n nodes has throttling number n if and only if it does not contain any 4-path, 4-cycle, or bowtie graph as an induced subgraph (Carlson and Kritschgau 2021, Theorem 4.2). Apparently, all these graphs can be constructed in the following way (for n >= 2). Let the nodes be 1, ..., n and choose a subset of special nodes. We require that n is a special node and that 1 is not, so there are 2^(n-2) possible choices for the set of special nodes. For i < j, let there be an edge between i and j if and only if j is a special node. These graphs do not contain any of the forbidden induced subgraphs, and different sets of special nodes lead to nonisomorphic graphs. Consequently, T(n,n) >= 2^(n-2). Apparently, equality holds, so that there are no other such graphs.
%H Steve Butler and Michael Young, <a href="https://orion.math.iastate.edu/butler/papers/13_02_throttling.pdf">Throttling zero forcing propagation speed on graphs</a>, Australasian Journal of Combinatorics 57 (2013), 65-71.
%H Joshua Carlson and Juergen Kritschgau, <a href="https://arxiv.org/abs/1909.07952">Various characterizations of throttling numbers</a>, arXiv:1909.07952 [math.CO], 2021.
%F T(n,k) = 0 for k < A000267(n-1), and T(n,A000267(n-1)) > 0 because the n-path has throttling number A000267(n-1) (Butler and Young 2013).
%e Triangle begins:
%e n\k 1 2 3 4 5 6 7 8 9 10
%e -----------------------------------------------------------
%e 1: 1
%e 2: 0 1
%e 3: 0 0 2
%e 4: 0 0 2 4
%e 5: 0 0 0 13 8
%e 6: 0 0 0 13 83 16
%e 7: 0 0 0 0 308 513 32
%e 8: 0 0 0 0 296 7460 3297 64
%e 9: 0 0 0 0 68 38630 200207 22047 128
%e 10: 0 0 0 0 0 65211 4788564 6709094 153446 256
%Y Row sums: A001349.
%Y Cf. A000267, A343648.
%K nonn,tabl
%O 1,6
%A _Pontus von Brömssen_, Apr 24 2021