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  • On singular signed graphs with nullspace spanned by a full vector : signed nut graphs
    Bašić, Nino ...
    A signed graph has edge weights drawn from the set ▫$\{+1,-1\}$▫, and is \emph{sign-balanced} if it is equivalent to an unsigned graph under the operation of sign switching; otherwise it is ... \emph{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 ▫$\rho \le 11$▫. 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. Orders for which a regular nut graph with ▫$\rho = n-1$▫ exists are characterised; they are \emph{sign-unbalanced} with an underlying graph ▫$K_n$▫ for which ▫$n \equiv 1 \pmod 4$▫. Orders for which a regular \emph{sign-unbalanced} nut graph with ▫$\rho = n - 2$▫ exists are also characterised; they have an underlying cocktail-party graph ▫$\mathrm{CP} (n)$▫ with even order ▫$n \geq 8$▫.
    Vir: Discussiones mathematicae. Graph theory. - ISSN 1234-3099 (Vol. 42, no. 4, 2022, str. 1351-1382)
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2022
    Jezik - angleški
    COBISS.SI-ID - 82446339
    DOI

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