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

From the theory of Ho0man polynomial, it is known that the adjacency matrix A of a strongly connected regular digraph of order n satis3es certain polynomial equation AP(A)=Jn, where l is a nonnegative integer, P(x) is a polynomial with rational coe5cients, and Jn is the n×n matrix of all ones. In this paper we present some su5cient conditions, in terms of the coe5cients 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, (A), is the lth iterated line digraph of (C). As a result, we simplify the study of some digraph classes with order functions asymptotically attaining the Moore bound. ? 2002 Elsevier Science B.V. All rights reserved.