Abstract
Let X be a graph on ν vertices with adjacency matrix A, and let S be a subset of its vertices with characteristic vector z. We say that the pair (X, S) is controllable if the vectors A r z for r = 1, . . . , ν − 1 span \({\mathbb{R}^{\nu}}\) . Our concern is chiefly with the cases where S = V(X), or S is a single vertex. In this paper we develop the basic theory of controllable pairs. We will see that if (X, S) is controllable then the only automorphism of X that fixes S as a set is the identity. If (X, S) is controllable for some subset S then the eigenvalues of A are all simple.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Godsil, C., Severini, S.: Control by quantum dynamics on graphs. Phys. Rev. A (3) 81(5), #Q052316, (2010)
Godsil C.D, McKay B.D: Spectral conditions for the reconstructibility of a graph. J. Combin. Theory Ser. B 30(3), 285–289 (1981)
Johnson C.R., Newman M.: A note on cospectral graphs. J. Combin. Theory Ser. B 28(1), 96–103 (1980)
Kailath, T.: Linear Systems. Prentice-Hall Inc., Englewood Cliffs, N.J. (1980)
Wang W., Xu C.X.: A sufficient condition for a family of graphs being determined by their generalized spectra. European J. Combin. 27(6), 826–840 (2006)