A note on Cayley nut graphs whose degree is divisible by four

Abstract

A nut graph is a non-trivial simple graph such that its adjacency matrix has a one-dimensional null space spanned by a full vector. It was recently shown by the authors that there exists a $d$-regular circulant nut graph of order $n$ if and only if $4\mid d,2\mid n,d>0$, together with $n\ge d+4$ if $d{\equiv }_{8}4$ and $n\ge d+6$ if $8\mid d$, as well as $(n,d)\ne (16,8)$ [arXiv:2212.03026, 2022]. In this paper, we demonstrate the existence of a $d$-regular Cayley nut graph of order $n$ for each $4\mid d,d>0$ and $2\mid n,n\ge d+4$, thereby resolving the existence problem for Cayley nut graphs and vertex-transitive nut graphs whose degree is divisible by four.