On $12$-regular nut graphs

TL;DR

This paper determines all possible orders for 12-regular nut graphs and explores the existence of circulant nut graphs for degrees divisible by 4, extending previous classifications for degrees 3 to 11.

Contribution

It completes the classification of regular nut graphs by identifying all orders for 12-regular nut graphs and analyzes circulant nut graphs for degrees divisible by 4.

Findings

All values of n for which 12-regular nut graphs exist are identified.

Infinitely many circulant nut graphs of degree divisible by 4 exist.

No circulant nut graphs exist for degrees congruent to 2 mod 4.

Abstract

A nut graph is a simple graph whose adjacency matrix is singular with $1$-dimensional kernel such that the corresponding eigenvector has no zero entries. In 2020, Fowler et al. characterised for each $d \in \{3,4,\ldots,11\}$ all values $n$ such that there exists a $d$-regular nut graph of order $n$. In the present paper, we determine all values $n$ for which a $12$-regular nut graph of order $n$ exists. We also present a result by which there are infinitely many circulant nut graphs of degree $d \equiv 0 \pmod 4$ and no circulant nut graph of degree $d \equiv 2 \pmod 4$.