On central digraphs constructed from left loops and loops

Abstract

Let An be the set of n X n zero-one matrices satisfying the matrix equation A2 = Jn; where Jn is n X n matrices of all ones. In this article, it is proved that the number of non-isomorphic left loops of order k gives the lower bound to the size of An for n = k2. Mainly we have established that any matrix in An corresponding to loop has rank 2k - 2, where n = k2, for some positive integer k.