Complete resolution of the circulant nut graph order-degree existence problem

TL;DR

This paper completely solves the problem of determining all pairs of order and degree for which circulant nut graphs exist, building on and extending previous partial results in the field.

Contribution

It provides a full characterization of all possible order-degree pairs for circulant nut graphs, resolving the longstanding existence problem.

Findings

All pairs (n, d) for circulant nut graphs are now known.

The results extend previous partial existence theorems.

The paper confirms the existence of such graphs for all feasible pairs.

Abstract

A circulant nut graph is a non-trivial simple graph such that its adjacency matrix is a circulant matrix whose null space is spanned by a single vector without zero elements. Regarding these graphs, the order-degree existence problem can be thought of as the mathematical problem of determining all the possible pairs $(n, d)$ for which there exists a $d$-regular circulant nut graph of order $n$. This problem was initiated by Ba\v{s}i\'c et al. and the first major results were obtained by Damnjanovi\'c and Stevanovi\'c, who proved that for each odd $t \ge 3$ such that $t\not\equiv_{10}1$ and $t\not\equiv_{18}15$, there exists a $4t$-regular circulant nut graph of order $n$ for each even $n \ge 4t + 4$. Afterwards, Damnjanovi\'c improved these results by showing that there necessarily exists a $4t$-regular circulant nut graph of order $n$ whenever $t$ is odd, $n$ is even, and $n \ge 4t + 4$ holds, or whenever $t$ is even, $n$ is such that $n \equiv_4 2$, and $n \ge 4t + 6$ holds. In this paper, we extend the aforementioned results by completely resolving the circulant nut graph order-degree existence problem. In other words, we fully determine all the possible pairs $(n, d)$ for which there exists a $d$-regular circulant nut graph of order $n$.