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

Abstract

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 A l P ( A )= J n , where l is a nonnegative integer, P ( x ) is a polynomial with rational coefficients, and J n 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 )= J n / d l 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.