On the Perron root and eigenvectors of a non-negative integer matrix

In this paper, we obtain a combinatorial expression for the Perron root and eigenvectors of a non-negative integer matrix using techniques from symbolic dynamics. We associate such a matrix with a multigraph and consider the edge shift corresponding to it. This gives rise to a collection of forbidden words $F$ which correspond to the non-existence of an edge between two vertices, and a collection of repeated words $R$ with multiplicities which correspond to multiple edges between two vertices. In general, for given collections $F$ of forbidden words and $R$ of repeated words with pre-assigned multiplicities, we construct a generalized language as a multiset. A combinatorial expression that enumerates the number of words of fixed length in this generalized language gives the Perron root and eigenvectors of the adjacency matrix. We also obtain conditions under which such a generalized language is a language of an edge shift.