TOPICS
An almost controllable graph is a graph on 
vertices whose walk
matrix has matrix rank 
(Wang et al. 2021, Wang and Wang 2025). 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.
Main Eigenvalues." Disc. Math. 344, 112397, 2021.Du,
Z., You, L., and Liu, H. "Almost Controllable Graphs and Beyond." Disc.
Math. 347, 113743, 2024.Du, Z.; You, L.; Liu, H.; and Liu,
F. "Further Results on Almost Controllable Graphs." Linear Algebra Appl. 677,
31-50, 2023.Liu, F. and Siemons, J. "Unlocking the Walk Matrix
of a Graph." J. Algebraic Combin. 55, 663-69, 2022.Qiu,
L.; Wang, W.; Wang, W.; and Zhang, H. "A New Criterion for Almost Controllable
Graphs Being Determined by Their Generalized Spectra." Disc. Math. 345,
113060, 2022.Sloane, N. J. A. Sequence A371919
in "The On-Line Encyclopedia of Integer Sequences."Wang, W.;
Liu, F.; and Wang, W. "Generalized Spectral Characterizations of Almost Controllable
Graphs." Europ. J. Combin. 96, 103348, 2021.Wang,
W. and Wang, W. "Haemers' Conjecture: An Algorithmic Perspective." Experimental
Math. 34, 147-161, 2025. https://doi.org/10.1080/10586458.2024.2337229.Weisstein, Eric W. "Almost Controllable Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlmostControllableGraph.html
