On cubic polycirculant nut graphs

TL;DR

This paper investigates the existence of cubic $ ext{ell}$-circulant nut graphs for various $ ext{ell}$ values, proving nonexistence for some and constructing infinitely many for others, advancing understanding of these specialized graphs.

Contribution

It proves the nonexistence of cubic 4- and 5-circulant nut graphs and constructs infinitely many for certain other $ ext{ell}$ values, extending previous classifications.

Findings

No cubic 4-circulant nut graphs exist.

No cubic 5-circulant nut graphs exist.

Infinitely many cubic $ ext{ell}$-circulant nut graphs exist for $ ext{ell} ext{ in } igrace{6,7igrace} ext{ or } ext{ell} ext{ at least } 9.

Abstract

A nut graph is a nontrivial simple graph whose adjacency matrix contains a one-dimensional null space spanned by a vector without zero entries. Moreover, an $\ell$-circulant graph is a graph that admits a cyclic group of automorphisms having $\ell$ vertex orbits of equal size. It is not difficult to observe that there exists no cubic $1$-circulant nut graph or cubic $2$-circulant nut graph, while the full classification of all the cubic $3$-circulant nut graphs was recently obtained [Electron. J. Comb. 31(2) (2024), #2.31]. Here, we investigate the existence of cubic $\ell$-circulant nut graphs for $\ell \ge 4$ and show that there is no cubic $4$-circulant nut graph or cubic $5$-circulant nut graph by using a computer-assisted proof. Furthermore, we rely on a construction based approach in order to demonstrate that there exist infinitely many cubic $\ell$-circulant nut graphs for any fixed $\ell \in \{6, 7 \}$ or $\ell \ge 9$.