Regular Steinhaus graphs of odd degree

Abstract

A Steinhaus matrix is a binary square matrix of size n which is symmetric, with a diagonal of zeros, and whose upper-triangular coefficients satisfy a i , j = a i − 1 , j − 1 + a i − 1 , j for all 2 ⩽ i < j ⩽ 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 K 2 is the only regular Steinhaus graph of odd degree. Using Dymacek’s theorem, we prove that if ( a i , j ) 1 ⩽ i , j ⩽ n is a Steinhaus matrix associated with a regular Steinhaus graph of odd degree then its sub-matrix ( a i , j ) 2 ⩽ i , j ⩽ n − 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 all even numbers n , and on ⌈ n 30 ⌉ parameters in the odd case. This result permits us to verify Dymacek’s conjecture up to 1500 vertices in the odd case.