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Nut digraphs

Nino Bašić, Patrick W. Fowler, Maxine M. McCarthy, Primož Potočnik

TL;DR

This work extends the concept of nut graphs from undirected to directed graphs by defining and studying five nut-digraph notions—$\text{dextro-nut}$, $\text{laevo-nut}$, $\text{bi-nut}$, $\text{ambi-nut}$, and $\text{inter-nut}$—with the kernel and cokernel framed via the adjacency matrix $A(G)$. It establishes core properties (strong connectivity, non-bipartiteness, degree constraints) for ambi-nut digraphs and develops a robust toolbox of constructions (subdivision, coalescence, cross-over, and gadgets/multiplier methods) to generate large families of ambi-nut graphs from smaller seeds, including infinite families like $M_k(n)$ and $D_k(n)$. The paper also surveys small examples, enumerations, and underlying graphs (notably circulants and Rose Window families), and discusses core-forbidden notions and inter-nut variants to illuminate the structure of singular directed graphs. Finally, it highlights potential applications in topological physics and molecular conduction, where such directed nullspaces can model directional transport and phase boundaries in non-Hermitian settings.

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, cross-over and multiplier constructions are introduced. Relevance of nut digraphs to topological physics is discussed.

Nut digraphs

TL;DR

This work extends the concept of nut graphs from undirected to directed graphs by defining and studying five nut-digraph notions—, , , , and —with the kernel and cokernel framed via the adjacency matrix . It establishes core properties (strong connectivity, non-bipartiteness, degree constraints) for ambi-nut digraphs and develops a robust toolbox of constructions (subdivision, coalescence, cross-over, and gadgets/multiplier methods) to generate large families of ambi-nut graphs from smaller seeds, including infinite families like and . The paper also surveys small examples, enumerations, and underlying graphs (notably circulants and Rose Window families), and discusses core-forbidden notions and inter-nut variants to illuminate the structure of singular directed graphs. Finally, it highlights potential applications in topological physics and molecular conduction, where such directed nullspaces can model directional transport and phase boundaries in non-Hermitian settings.

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 . Refined notions of core and core-forbidden vertices apply to singular digraphs. Infinite families of nut digraphs and systematic coalescence, cross-over and multiplier constructions are introduced. Relevance of nut digraphs to topological physics is discussed.

Paper Structure

This paper contains 13 sections, 29 theorems, 37 equations, 17 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Definitions
  3. Examples and enumeration
  4. Basic properties of nut digraphs
  5. Connectedness and vertex deletion
  6. Bipartiteness
  7. Vertex degrees and leaflessness
  8. Constructions
  9. Subdivision construction
  10. Coalescence construction
  11. Cross-over construction
  12. Multiplier constructions
  13. Perspective

Key Result

Proposition 6

The digraphs $M_1(n)$ and $M_2(n)$ are ambi-nut digraphs for every even $n \geq 6$. The graph $M_3(n)$ is an ambi-nut digraph for every even $n \geq 6$ that satisfies $n \not\equiv 0 \pmod{6}$.

Figures (17)

  • Figure 1: Local conditions in nut digraphs. Entries on $G^+(v)$ of $\mathbf{x} \in \mathop{\mathrm{ker}}\nolimits G$ are labeled $a_1, \ldots, a_k$ and entries on $G^-(v)$ of $\mathbf{y} \in \mathop{\mathrm{coker}}\nolimits G$ are labeled $b_1, \ldots, b_\ell$. For a dextro-nut digraph, the local condition is $a_1 + \cdots + a_k = 0$ and for a laevo-nut $b_1 + \cdots + b_\ell = 0$. For the underlying graph the local condition would simply be the sum of the two: $a_1 + \cdots + a_k + b_1 + \cdots + b_\ell = 0$.
  • Figure 2: Venn diagram of relationships amongst notions of nut digraphs. Symbols $\mathcal{D}, \mathcal{L}, \mathcal{B}, \mathcal{A}$ and $\mathcal{I}$ denote dextro-, laevo-, bi-, ambi- and inter-nut digraphs, respectively. Note that the usual nut graphs occupy the small red oval inside the ambi-nut region, and that the hatched regions are empty.
  • Figure 3: Examples of nut digraphs among oriented graphs: (a) the smallest dextro-nut digraph, (b) a dextro-nut digraph with a leaf, (c) & (d) the two smallest ambi-nut digraphs (labeled $M_1(6)$ and $M_2(6)$), (e) the unique ambi-nut digraph on 7 vertices, (f) one of the 20 bi-nut digraphs on 7 vertices that are not ambi-nuts. In all panels, vertex labels show the kernel vector. In case (f), the vectors from the kernel and co-kernel differ in just one entry (labeled $\mp 1$, with $-1$ corresponding to the kernel vector).
  • Figure 4: Small $4$-regular nut digraphs. Panels (a) & (b) show the two $4$-regular dextro-nut digraphs on $6$ vertices that are not bi-nut digraphs. Both have the same underlying graph as $M_1(6)$, and since each of them contains a sink, they are not isomorphic to $M_1(6)$ or $M_2(6)$. Panels (c) to (g) show the full set of five $4$-regular ambi-nut digraphs on $8$ vertices.
  • Figure 5: Small $4$-regular ambi-nut digraphs with a Rose Window graph as underlying graph.
  • ...and 12 more figures

Theorems & Definitions (66)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 6
  • proof
  • Definition 7
  • Proposition 8
  • proof
  • ...and 56 more