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Classification of quartic bicirculant nut graphs

Ivan Damnjanović, Nino Bašić, Tomaž Pisanski, Arjana Žitnik

TL;DR

The paper resolves the quartic bicirculant nut graph problem by deriving a complete classification across the four connected quartic bicirculant classes. It leverages root-of-unity criteria, explicit bicirculant eigenvalue formulas, and cyclotomic divisibility results to obtain necessary and sufficient conditions for nuttiness, with detailed theorems for classes $\\mathcal{B}_1, \\mathcal{B}_2, \\mathcal{B}_3$ and a remark that $\\mathcal{B}_4$ is excluded by bipartiteness. The work combines rigorous algebraic techniques with computational checks to enumerate and exemplify the nut graphs up to order $50$, including special cases like Rose Window graphs and various Cayley/Circulant instances. The results advance understanding of how symmetry and spectral properties interact in structured graph families, and lay groundwork for future study of automorphism groups and class overlaps among quartic bicirculants.

Abstract

A graph is called a nut graph if zero is its eigenvalue of multiplicity one and its corresponding eigenvector has no zero entries. A graph is a bicirculant if it admits an automorphism with two equally sized vertex orbits. There are four classes of connected quartic bicirculant graphs. We classify the quartic bicirculant graphs that are nut graphs by investigating properties of each of these four classes.

Classification of quartic bicirculant nut graphs

TL;DR

The paper resolves the quartic bicirculant nut graph problem by deriving a complete classification across the four connected quartic bicirculant classes. It leverages root-of-unity criteria, explicit bicirculant eigenvalue formulas, and cyclotomic divisibility results to obtain necessary and sufficient conditions for nuttiness, with detailed theorems for classes and a remark that is excluded by bipartiteness. The work combines rigorous algebraic techniques with computational checks to enumerate and exemplify the nut graphs up to order , including special cases like Rose Window graphs and various Cayley/Circulant instances. The results advance understanding of how symmetry and spectral properties interact in structured graph families, and lay groundwork for future study of automorphism groups and class overlaps among quartic bicirculants.

Abstract

A graph is called a nut graph if zero is its eigenvalue of multiplicity one and its corresponding eigenvector has no zero entries. A graph is a bicirculant if it admits an automorphism with two equally sized vertex orbits. There are four classes of connected quartic bicirculant graphs. We classify the quartic bicirculant graphs that are nut graphs by investigating properties of each of these four classes.

Paper Structure

This paper contains 10 sections, 25 theorems, 47 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic results about nut graphs
  4. Eigenvalues of bicirculants
  5. Cyclotomic polynomials
  6. The class B1
  7. The class B2 --- The generalized rose window graphs
  8. The class B3
  9. Conclusion
  10. SageMath script for Theorem \ref{['b2_main_th']}

Key Result

Theorem 1.1

A quartic bicirculant graph is a nut graph if and only if it is isomorphic to one of the graphs below: In particular, no graphs from the class $\mathcal{B}_4$ are nut graphs. In conditions 2(iii) and 2(iv), the $\pm$ signs can be chosen independently.

Figures (2)

  • Figure 1: Possible $\mathbb{Z}_m$-voltage graphs for the connected quartic bicirculants. Cases (1) and (3) occur only when $m$ is even and in this case, the voltages on the semi-edges are equal to $\frac{m}{2}$.
  • Figure 2: The smallest examples of QBN graphs. Kernel eigenvector entries are color-coded: one color represents entries $+1$, while the other represents $-1$.

Theorems & Definitions (49)

  • Theorem 1.1: Quartic bicirculant nut graph classification
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Remark
  • Corollary 2.6
  • ...and 39 more