A Matrix for Counting Paths in Acyclic Digraphs

We will define a matrix A=A1 associated with an acyclic digraph 1, such that the coefficients of the characteristic polynomial of A enumerate the paths in 1 according to their length. Our result is actually an easy consequence of a more general theorem of Goulden and Jackson, but this special case seems never to have been explicitly noted before. We will give two simple proofs, the first of which is essentially a specialization of the proof of Goulden and Jackson. Let 1 be an acyclic digraph without multiple edges on the vertex set V=[x1 , ..., xn]. We say that 1 is natural if i< j whenever (xi , xj) is an edge. We may assume without loss of generality throughout this paper that 1 is natural. Regard x1 , ..., xn as indeterminates, and define the diagonal matrix D=diag(x1 , ..., xn). Define the n_n matrix A=A1 by Aij={0, 1, if (xi , xj) is an edge of 1 otherwise. (1)