Geodesic
On Singular Signed Graphs with Nullspace Spanned by a Full Vector: Signed Nut Graphs
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 4, pp. 1351-1382

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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 (ρ, n) for which a ρ-regular nut graph (sign-balanced or sign-unbalanced) of order n exists with ρ ≤ 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 ρ -regular nut graphs. Orders for which a regular nut graph with ρ = n − 1 exists are characterised; they are sign-unbalanced with an underlying graph Kn for which n ≡ 1 (mod 4). Orders for which a regular sign-unbalanced nut graph with ρ = n − 2 exists are also characterised; they have an underlying cocktail-party graph CP(n) with even order n ≥ 8.
Keywords: signed graph, nut graph, singular graph, graph spectrum, Fowler construction, sign-balanced graph, sign-unbalanced graph, cocktail-party graph 
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     title = {On {Singular} {Signed} {Graphs} with {Nullspace} {Spanned} by a {Full} {Vector:} {Signed} {Nut} {Graphs}},
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Bašić, Nino; Fowler, Patrick W.; Pisanski, Tomaž; Sciriha, Irene. On Singular Signed Graphs with Nullspace Spanned by a Full Vector: Signed Nut Graphs. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 4, pp. 1351-1382. http://geodesic.mathdoc.fr/item/DMGT_2022_42_4_a18/