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A circulant graph is a simple graph whose adjacency matrix can be representedin the form of a circulant matrix, while a nut graph is considered to be agraph whose null space is spanned by a single full vector. In a previous studyby Damnjanovi\'c [arXiv:2212.03026, 2022], the complete set of all the pairs$(n, d)$ for which there exists a $d$-regular circulant nut graph of order $n$has been determined. Motivated by the said results, we put our focus on thequartic circulant graphs and derive an explicit formula for computing theirnullities. Furthermore, we implement the aforementioned formula in order toobtain a method for inspecting the singularity of a particular quarticcirculant graph and find the concise criteria to be used for testing whethersuch a graph is a nut graph. Subsequently, we compute the minimum and maximumnullity that a quartic circulant graph of a fixed order $n$ can attain, foreach viable order $n \ge 5$. Finally, we determine all the graphs attainingthese nullities and then provide a full characterization of all of theircorresponding extremal null spaces.

On the nullities of quartic circulant graphs and their extremal null spaces