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Nut Combinatorics Signed graph Indifference graph Chordal graph Sign (mathematics) Mathematics Discrete mathematics Graph Physics Acoustics Mathematical analysis

Nino Bašić , Patrick W. Fowler , Tomaž Pisanski , Irene Sciriha ·

YOU? · · 2020 · Open Access · · OA: W3087605174

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 sign-balanced if it is equivalent to an unsigned graph under an intuitive operation of sign switching, otherwise it is sign-unbalanced. By including signed nut graphs, we find all pairs $(\rho, n)$ for which a $\rho$-regular nut graph of order $n$ exists with $\rho \le 11$. In addition, we show how a literature construction for obtaining larger nut graphs can be extended to signed graphs, giving a construction for both sign-balanced and sign-unbalanced $\rho$-regular signed nut graphs.