On Singular Signed Graphs with Nullspace Spanned by a Full Vector: Signed Nut Graphs

A signed graph has edge weights drawn from the set {+1, -1}, and is sign-balanced if it is equivalent to an unsigned graph under the operation of sign switching; otherwise it is sign-unbalanced. A nut graph has a one dimensional kernel of the 0-1 adjacency matrix with a corresponding eigenvector that is full. In this paper we generalise the notion of nut graphs to signed graphs. Orders for which regular nut graphs with all edge weights +1 exist have been determined recently for the degrees up to 12. By extending the definition to signed graphs, we here find all pairs (rho, n) for which a rho-regular nut graph (sign-balanced or sign-unbalanced) of order n exists with rho <= 11. We devise a construction for signed nut graphs based on a smaller 'seed' graph, giving infinite series of both sign-balanced and sign-unbalanced rho -regular nut graphs. Orders for which a regular nut graph with rho = n - 1 exists are characterised; they are sign-unbalanced with an underlying graph K-n for which n equivalent to 1 (mod 4). Orders for which a regular sign-unbalanced nut graph with rho = n - 2 exists are also characterised; they have an underlying cocktail-party graph CP(n) with even order n >= 8.