Define the walk matrix of a graph on vertices with adjacency matrix
as
where
is an -vector
consisting of all 1's. A graph with vertex count for which the matrix rank of
its walk matrix equals is then said to be almost controllable (Wang et al. 2021,
Wang and Wang 2024). The numbers of almost controllable simple graphs on , 2, ... vertices are 0, 2, 2, 2, 6, 22, 214, 3100, 86578,
3712582, ... (OEIS A371919).
The first few almost controllable graphs are illustrated above.
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