Vertex-transitive nut graph order-degree existence problem

TL;DR

This paper proves the existence of vertex-transitive nut graphs with specified order and degree, specifically for all pairs satisfying certain divisibility and inequality conditions, expanding the known classes of such graphs.

Contribution

It establishes the existence of d-regular Cayley nut graphs for all pairs (n, d) meeting specific divisibility and size criteria, filling a gap in the graph theory literature.

Findings

Existence of d-regular Cayley nut graphs for all qualifying (n, d) pairs.

Complete characterization of (n, d) pairs for vertex-transitive nut graphs.

Provides constructions for these graphs based on group theory.

Abstract

A nut graph is a nontrivial simple graph whose adjacency matrix has a simple eigenvalue zero such that the corresponding eigenvector has no zero entries. It is known that the order $n$ and degree $d$ of a vertex-transitive nut graph satisfy $4 \mid d$, $d \ge 4$, $2 \mid n$ and $n \ge d + 4$; or $d \equiv 2 \pmod 4$, $d \ge 6$, $4 \mid n$ and $n \ge d + 6$. Here, we prove that for each such $n$ and $d$, there exists a $d$-regular Cayley nut graph of order $n$. As a direct consequence, we obtain all the pairs $(n, d)$ for which there is a $d$-regular vertex-transitive (resp. Cayley) nut graph of order $n$.