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# Controllable Graph

Define the walk matrix of a graph on vertices with adjacency matrix as

where is an -vector consisting of all 1's. Then a graph is said to be controlable if its walk matrix is invertible (Godsil 2012). Equivalently, graph is controllable iff the matrix rank of its walk matrix equals the graph's vertex count .

Controllable graphs are identity graphs (Godsil 2012).

A graph is controllable iff its graph complement is (Godsil 2012).

O'Rourke and Touri (2016) proved that almost all graphs are controllable (Wang and Wang 2024). The numbers of controllable graphs on , 2, ... vertices are 1, 0, 0, 0, 0, 8, 92, 2332, 85036, 5578994, ... (OEIS A356669; Farrugia 2016) (summarized in the following table) and the corresponding numbers of connected controllable graphs are 1, 0, 0, 0, 0, 8, 85, 2275, 83034, 5512362, ... (OEIS A371897).

 # controllable # graphs fraction OEIS A356669 A00088 1 1 1 100% 2 0 2 0% 3 0 4 0% 4 0 11 0% 5 0 34 0% 6 8 156 5.13% 7 92 1044 8.81% 8 2332 12346 18.9% 9 85036 274668 31.0% 10 5578994 12005168 46.5%

The eight controllable graphs on 6 vertices are illustrated above.

A graph with vertex count for which the matrix rank of its walk matrix equals is said to be almost controllable (Wang et al. 2021, Wang and Wang 2024).