Search the repository

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 (Damnjanović et al. in Electron. J. Comb. 31(2):P2.31, 2024). Here, we investigate the existence of cubic $\ell$-circulant nut graphs for $\ell \geq 4$ 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 $\ell \geq 9$.
Keywords: nut graph, polycirculant graph, cubic graph, pregraph, voltage graph
Published in RUP: 19.11.2025; Views: 307; Downloads: 5
Full text (581,83 KB)
This document has more files! More...