  Call two pairs $(M,N)$ and $(M',N')$ of $m\times n$ matrices over a field
$K$, \emph{simultaneously K-equivalent} if there exist square invertible
matrices $S,T$ over K, with $M'=SMT$ and $N'=SNT$. Kronecker \cite{Kronecker}
has given a complete set of invariants for simultaneous equivalence of pairs of
matrices.
  Associate in the natural way to a finite directed graph $\Gamma$, with $v$
vertices and $e$ edges, an ordered pair $(M,N)$ of $e\times v$ matrices of
zeros and ones. It is natural to try to compute the Kronecker invariants of
such a pair $(M,N)$, particularly since they clearly furnish
isomorphism-invariants of $\Gamma$. Let us call two graphs `linearly
equivalent' when their two corresponding pairs are simultaneously equivalent.
  The purpose of the present paper, is to compute directly these Kronecker
invariants of finite directed graphs, from elementary combinatorial properties
of the graphs. A pleasant surprise is that these new invariants are purely {\bf
rational} --indeed, {\bf integral}, in the sense that the computation needed to
decide if two directed graphs are linearly equivalent only involves counting
vertices in various finite graphs constructed from each of the given graphs--
and {\bf does not involve finding the irreducible factorization of a polynomial
over K} (in apparent contrast both to the familiar invariant-computations of
graphs furnished by the eigenvalues of the connection matrix, and to the
isomorphism problem for general pairs of matrices.)
