Nut graphs with a prescribed number of vertex and edge orbits

TL;DR

This paper characterizes the existence of nut graphs with specified numbers of vertex and edge orbits, extending previous results and constructing infinite families with particular symmetry properties.

Contribution

It determines all pairs (r, k) for which nut graphs with r vertex orbits and k edge orbits exist, including infinite families of Cayley nut graphs.

Findings

Existence of nut graphs for all pairs (r, k) with certain conditions.

Infinite families of Cayley nut graphs with equal edge and arc orbits.

Extension of previous results on the symmetry properties of nut graphs.

Abstract

A nut graph is a nontrivial graph whose adjacency matrix has a one-dimensional null space spanned by a vector without zero entries. Recently, it was shown that a nut graph has more edge orbits than vertex orbits. It was also shown that for any even $r \ge 2$ and any $k \ge r + 1$, there exist infinitely many nut graphs with $r$ vertex orbits and $k$ edge orbits. Here, we extend this result by finding all the pairs $(r, k)$ for which there exists a nut graph with $r$ vertex orbits and $k$ edge orbits. In particular, we show that for any $k \ge 2$, there are infinitely many Cayley nut graphs with $k$ edge orbits and $k$ arc orbits.