On circulant nut graphs

TL;DR

This paper investigates circulant nut graphs, characterizing their generator sets, resolving conjectures about their existence for various orders and degrees, and expanding understanding of their structural properties.

Contribution

It provides a complete characterization of circulant nut graphs with specific generator sets and resolves key conjectures about their existence for various parameters.

Findings

Generator set of a circulant nut graph contains equal numbers of even and odd integers.

Characterization of circulant nut graphs with generator set \\{x, x+1, ..., x+2t-1\\}.

Existence of even order circulant nut graphs for certain generator sets and parameters.

Abstract

A nut graph is a simple graph whose adjacency matrix has the eigenvalue~0 with multiplicity~1 such that its corresponding eigenvector has no zero entries. Motivated by a question of Fowler et al.~[\emph{Disc. Math. Graph Theory} 40 (2020), 533--557] to determine the pairs $(n,d)$ for which a vertex-transitive nut graph of order $n$ and degree $d$ exists, Ba\v si\'c et al.\ [\arxiv{2102.04418}, 2021] initiated the study of circulant nut graphs. Here we first show that the generator set of a circulant nut graph necessarily contains equally many even and odd integers. Then we characterize circulant nut graphs with the generator set $\{x,x+1,\dots,x+2t-1\}$ for $x,t\in\N$, which generalizes the result of Ba\v si\'c et al.\ for the generator set $\{1,\dots,2t\}$. We further study circulant nut graphs with the generator set $\{1,\dots,2t+1\}\setminus\{t\}$, which yields nut graphs of every even order $n\geq 4t+4$ whenever $t$~is odd such that $t\not\equiv_{10}1$ and $t\not\equiv_{18}15$. This fully resolves Conjecture~9 from Ba\v si\'c et al.~[ibid.]. We also study the existence of $4t$-regular circulant nut graphs for small values of~$t$, which partially resolves Conjecture~10 of Ba\v si\'c et al.~[ibid.].