On the matrix equation Ak=J−I Original Research Article

We concern ourselves with the problem of solving the (0,1) matrix equation Ak=J−I in this paper, where J is the matrix of all one’s, I the identity matrix and A an unknown (0,1) matrix. In particular, our effort brings about a complete solution of Ak=J2k+1−I2k+1. This generalizes a theorem of Lam and Van Lint. In the course of our solution we provide a local characterization of the webs, i.e., the powers of the cycles. Our results mainly rely on the analysis of the intersection pattern of a collection of some specific sets, namely, the row sets of a matrix; some results on partitionable graphs are also introduced to tackle this problem. Our work suggests an approach to investigate Ak=J−I by studying a number-theoretical question and a conjecture of Ravindra on partitionable graphs. Several open problems are also presented.