On the degrees of regular nut graphs and Cayley nut graphs

TL;DR

This paper proves the existence of infinitely many regular nut graphs for all degrees greater than or equal to 3, and specifically constructs infinitely many Cayley nut graphs for even degrees, clarifying feasible degrees for such graphs.

Contribution

It extends the known existence results of regular nut graphs to all degrees ≥3 and identifies all degrees for which Cayley nut graphs exist.

Findings

Existence of infinitely many d-regular nut graphs for all d ≥ 3.

Existence of infinitely many d-regular Cayley nut graphs for all even d ≥ 4.

Complete characterization of feasible degrees for Cayley nut graphs.

Abstract

A nut graph is a simple graph for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry. It is known that infinitely many $d$-regular nut graphs exist for $3 \leq d \leq 12$ and for $d \geq 4$ such that $d \equiv 0 \pmod{4}$. Here it is shown that infinitely many $d$-regular nut graphs exist for each degree $d \geq 3$. Moreover, we prove that there are infinitely many $d$-regular Cayley nut graphs for each even $d \ge 4$. This implies that we have identified all feasible degrees $d$ for which a $d$-regular Cayley nut graph exists.