  Characterizing graphs by their spectra is an important topic in spectral
graph theory, which has attracted a lot of attention of researchers in recent
years. It is generally very hard and challenging to show a given graph to be
determined by its spectrum. In Wang~[J. Combin. Theory, Ser. B, 122
(2017):438-451], the author gave a simple arithmetic condition for a family of
graphs being determined by their generalized spectra. However, the method
applies only to a family of the so called \emph{controllable graphs}; it fails
when the graphs are non-controllable.
  In this paper, we introduce a class of non-controllable graphs, called
\emph{almost controllable graphs}, and prove that, for any pair of almost
controllable graphs $G$ and $H$ that are generalized cospectral, there exist
exactly two rational orthogonal matrices $Q$ with constant row sums such that
$Q^{\rm T}A(G)Q=A(H)$, where $A(G)$ and $A(H)$ are the adjacency matrices of
$G$ and $H$, respectively. The main ingredient of the proof is a use of the
Binet-Cauchy formula. As an application, we obtain a simple criterion for an
almost controllable graph $G$ to be determined by its generalized spectrum,
which in some sense extends the corresponding result for controllable graphs.
