g-Circulant solutions to the (0,1) matrix equation Am=Jn Original Research Article

This paper discusses the structure of g-circulant solutions to Am=Jn, where A is an unknown (0,1) matrix and Jn is a matrix of order n with all entries equal to 1. Wang has made a conjecture on the form of all such solutions. Partially verifying his conjecture, we discover a close relationship among the Hall polynomial θA(x), the shifting parameter g, and the order n of any (0,1) g-circulant solution A to Am=Jn. As a consequence, all the g-circulant solutions to Am=Jn are completely determined in the case that n is a prime power. Moreover, in the case that the constant line sum r of A is square-free, all g-circulant solutions to Am=Jn are proved to be permutation similar to the adjacency matrix of the De Bruijn digraph B(r,m). Motivated by the current status of this subject, we identify all (0,1) g-circulant solutions to Am=Jn whose Hall polynomials have some specific properties and we further determine the possible values that the shifting parameter g of such solutions may take. The uniqueness of these solutions up to isomorphism is also investigated. Our paper is concluded with some open problems. In particular, we give the concept of standard factorization and conjecture that all factorizations of (xn−1)/(x−1) into a product of (0,1) polynomials must be standard and thus point out the close similarity between Wangʹs conjecture and a conjecture appearing in the study of perfect graph.