Signed nut graphs

Abstract

Orders for which regular nut graphs exist have been determined recently for the degrees up to 11. In this paper we extend the notion of nut graphs to signed graphs, i.e. graphs with edges weighted either by +1 or −1. A signed graph is proper if it is not equivalent to an unsigned graph under an intuitive operation of sign switching, otherwise it is traditional. By including signed nut graphs, we find all pairs (ρ,n) for which a ρ-regular nut graph of order n exists with ρ≤11. In addition, we show how a literature construction for obtaining larger nut graphs can be extended to signed graphs, giving a construction for both proper and traditional ρ-regular signed nut graphs.