1. Nut digraphsNino Bašić, Patrick W. Fowler, Maxine M. McCarthy, Primož Potočnik, 2026, original scientific article Abstract: A nut graph is a simple graph whose kernel is spanned by a single full vector (i.e., the adjacency matrix has a single zero eigenvalue and all non-zero kernel eigenvectors have no zero entry). We classify generalisations of nut graphs to nut digraphs: a digraph whose kernel (resp. co-kernel) is spanned by a full vector is dextro-nut (resp. laevo-nut); a bi-nut digraph is both laevo- and dextro-nut; an ambi-nut digraph is a bi-nut digraph where kernel and co-kernel are spanned by the same vector; a digraph is inter-nut if the intersection of the kernel and co-kernel is spanned by a full vector. It is known that a nut graph is connected, leafless and non-bipartite. It is shown here that an ambi-nut digraph is strongly connected, non-bipartite (i.e., has a non-bipartite underlying graph) and has minimum in-degree and minimum out-degree of at least 2. Refined notions of core and core-forbidden vertices apply to singular digraphs. Infinite families of nut digraphs and systematic coalescence, crossover and multiplier constructions are introduced. Relevance of nut digraphs to topological physics is discussed.
Keywords: nut graph, core graph, nullity, directed graph, nut digraph, dextro-nut, laevo-nut, bi-nut, ambi-nut, inter-nut, dextro-core vertex, laevo-core vertex, graph spectra
Published in RUP: 09.01.2026; Views: 689; Downloads: 43
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2. Minimal directed strongly regular Cayley graphs over generalized dicyclic groupsYueli Han, Lu Lu, 2025, original scientific article Abstract: Let G be a group with identity element 1, and let S be a subset of G \ {1}. The subset S is called minimal if ⟨S⟩ = G and there exists an element s ∈ S such that ⟨S \ {s, s−1}⟩ ≠ G. In this paper, we completely determine all directed strongly regular Cayley graphs Cay(G, S) for any generalized dicyclic group G, provided that S is a minimal subset of G.
Keywords: directed strongly regular graph, Cayley graph, generalized dicyclic group
Published in RUP: 21.10.2025; Views: 859; Downloads: 9
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4. Posplošitev Lijeve domneve in popolna klasifikacija cikličnih m-(D)CI-grup : magistrsko deloLuka Šinkovec, 2023, master's thesis Keywords: (un)directed Cayley graph, cyclic group, (un)directed circulant graph, Cayley isomorphism, (un)directed CI-graph, (D)CI-group, m-(D)CI-group, key, generalised multiplier
Published in RUP: 11.09.2023; Views: 2634; Downloads: 48
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6. A note on a conjecture on consistent cyclesŠtefko Miklavič, 2013, original scientific article Abstract: Let ▫$\Gamma$▫ denote a finite digraph and let ▫$G$▫ be a subgroup of its automorphism group. A directed cycle ▫$\vec{C}$▫ of▫ $\Gamma$▫ is called ▫$G$▫-consistent whenever there is an element of ▫$G$▫ whose restriction to▫ $\vec{C}$▫ is the 1-step rotation of ▫$\vec{C}$▫. In this short note we provea conjecture on ▫$G$▫-consistent directed cycles stated by Steve Wilson.
Keywords: graph theory, digraphs, consistent directed cycles
Published in RUP: 15.10.2013; Views: 4893; Downloads: 134
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