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 .

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).

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).

Farrugia, A. "On Strongly Asymmetric and Controllable Primitive Graphs." Disc. Appl. Math.211, 58-67, 2016.Godsil,
C. "Controllable Subsets in Graphs." Ann. Comb.16, 733-744,
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Controllable Random Graphs." SIAM J. Control Optim.54, 3347-3378,
2016.Sloane, N. J. A. Sequences A356669
and A371897 in "The On-Line Encyclopedia
of Integer Sequences."Wang, W. and Wang, W. "Haemers' Conjecture:
An Algorithmic Perspective." Experimental Math., 10 Apr 2024.Wang,
W.; Liu, F.; and Wang, W. "Generalized Spectral Characterizations of Almost
Controllable Graphs." Europ. J. Combin.96, 103348, 2021.Yoon,
M.-G.; Rowlinson, P.; Cvetković, D.; and Stanić, Z. "Controllability
of Multi-Agent Dynamical Systems With a Broadcasting Control Signal." Asian
J. Control16, 1066-1072, 2014.