Nullities for a class of 0-1 symmetric Toeplitz band matrices

Let $S(n,k)$ denote the $n \times n$ symmetric Toeplitz band matrix whose first $k$ superdiagonals and first $k$ subdiagonals have all entries $1$, and whose remaining entries are all $0$. For all $n > k >0$ with $k$ even, we give formulas for the nullity of $S(n,k)$. As an application, it is shown that over half of these matrices $S(n,k)$ are nonsingular. For the purpose of rapid computation, we devise an algorithm that quickly computes the nullity of $S(n,k)$ even for extremely large values of $n$ and $k$, when $k$ is even. The algorithm is based on a connection between the nullspace vectors of $S(n,k)$ and the cycles in a certain directed graph.