Regular Steinhaus graphs of odd degree
Résumé
A Steinhaus matrix is a binary square matrix of size $n$ which is symmetric, with diagonal of zeros, and whose upper-triangular coefficients satisfy $a_{i,j}=a_{i-1,j-1}+a_{i-1,j}$ for all $2 \leqslant i < j \leqslant n$. 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 $\left(a_{i,j}\right)_{1\leqslant i,j\leqslant n}$ is a Steinhaus matrix associated with a regular Steinhaus graph of odd degree then its sub-matrix $\left(a_{i,j}\right)_{2\leqslant i,j\leqslant n-1}$ 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 $\left\lceil \frac{n}{24}\right\rceil$ parameters for all even numbers $n$, and on $\left\lceil \frac{n}{30}\right\rceil$ parameters in the odd case. This result permits us to verify the Dymacek's conjecture up to $1500$ vertices in the odd case.
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