The underlying line digraph structure of some (0,1)-matrix equations

From the theory of Hoffman polynomial, it is known that the adjacency matrix A of a strongly connected regular digraph of order n satisfies certain polynomial equation AlP(A)=Jn, where l is a nonnegative integer, P(x) is a polynomial with rational coefficients, and Jn is the n×n matrix of all ones. In this paper we present some sufficient conditions, in terms of the coefficients of P(x), to ensure that all (0,1)-matrices satisfying such an equation with l>0 have an underlying line digraph structure, that is to say, for any solution A there exists a (0,1)-matrix C satisfying P(C)=Jn/dl and the associated (d-regular) digraph of A, Γ(A), is the lth iterated line digraph of Γ(C). As a result, we simplify the study of some digraph classes with order functions asymptotically attaining the Moore bound.