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Almost Controllable Graph


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

 W(G)=[1,A1,...,A^(n-1)1],

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

AlmostControllableGraphs

The first few almost controllable graphs are illustrated above.


See also

Controllable Graph

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References

Du, Z.; Liu, F.; Liu, S.; and Qin, Z. "Graphs With n-1 Main Eigenvalues." Disc. Math. 344, 112397, 2021.Du, Z.; You, L.; Liu, H.; and Liu, F. "Further Results on Almost Controllable Graphs." Linear Algebra Appl. 677, 31-50, 2023.Du, Z., You, L., and Liu, H. "Almost Controllable Graphs and Beyond." Disc. Math. 347, 113743, 2024.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. 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.

Cite this as:

Weisstein, Eric W. "Almost Controllable Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AlmostControllableGraph.html

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