Nullities for a class of 0–1 symmetric toeplitz band matrices

Let S(n, k) denote the n×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.