Regular Steinhaus graphs of odd degree

A Steinhaus matrix is a binary square matrix of size n which is symmetric, with diagonal of zeros, and whose upper-triangular coefficients satisfy ai,j = ai−1,j−1+ai−1,j for all 2 6 i < j 6 n. Steinhaus matrices are determined by their first row. A Steinhaus graph is a simple graph whose adjacency matrix is a Steinhaus matrix. We give a short new proof of a theorem, due to Dymacek, which states that even Steinhaus graphs, i.e. those with all vertex degrees even, have doubly-symmetric Steinhaus matrices. In 1979 Dymacek conjectured that the complete graph on two vertices K2 is the only regular Steinhaus graph of odd degree. Using Dymacek’s theorem, we prove that if (ai,j)16i,j6n is a Steinhaus matrix associated to a regular Steinhaus graph of odd degree then its sub-matrix (ai,j)26i,j6n−1 is a multi-symmetric matrix, that is a doubly-symmetric matrix where each row of its upper-triangular part is a symmetric sequence. We prove that the multi-symmetric Steinhaus matrices of size n whose Steinhaus graphs are regular modulo 4, i.e. where all vertex degrees are equal modulo 4, only depend on ⌈ n 24 ⌉ parameters for every even number n, and on ⌈ n 30 ⌉ parameters in the odd case. This result permits us to verify the Dymacek’s conjecture up to 1500 vertices in the odd case.