DIRECTED GRAPHS AND KRONECKER INVARIANTS OF PAIRS OF MATRICES

Abstract

Call two pairs (M,N) and (M′,N′) of m × n matrices over a field K, simultaneously K-equivalent if there exist square invertible matrices S,T over K, with M′ = SMT and N′ = SNT. Kronecker [2] has given a complete set of invariants for simultaneous equivalence of pairs of matrices. Associate in the natural way to a finite directed graph Γ, with v vertices and e edges, an ordered pair (M,N) of e × 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 Γ. Let us call two graphs "linearly equivalent" when their two corresponding pairs are simultaneously equivalent. There have existed, since 1890, highly effective algorithms for computing the Kronecker invariants of pairs of matrices of the same size over a given field [1,2,5,6] and in particular for those arising in the manner just described from finite directed graphs. 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 rational — indeed, 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 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).