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 termed sign-balanced if it is equivalent to an unsigned graph under the operation of sign switching; otherwise it is called sign-unbalanced. A nut graph has a one dimensional kernel with a corresponding eigenvector that is full. In this paper we generalise the notion of nut graphs to signed graphs. Orders for which unsigned regular nut graphs exist were determined recently for the degrees up to $11$. By extending the definition to signed nut graphs, we find all pairs $(\rho, n)$ for which a $\rho$-regular nut graph (sign-balanced or sign-unbalanced) of order $n$ exists with $\rho \le 11$. 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. All orders for which a complete sign-unbalanced nut graph exists are characterised; they have underlying graph $K_n$ with $n \equiv 1 \pmod 4$. All orders for which a regular sign-unbalanced nut graph with $\rho = n - 2$ exists are also characterised; they have an underlying cocktail-party graph $\mathrm{CP}(n)$ with even $n \geq 8$.