On the null spaces of quartic circulant graphs

A nut graph is a nontrivial simple graph whose adjacency matrix has a one-dimensional null space such that its nonzero vectors contain no zero elements. For circulant graphs, it is known that they are nut if and only if their nullity is one. This fact was recently used by the author in order to show that there exists a d-regular circulant nut graph of order n if and only if 4 ∣ d, 2 ∣ n, d > 0, alongside n ≥ d + 4 if d≡₈4, and n ≥ d + 6 if 8 ∣ d, and (n, d) ≠ (16, 8). Here, we deal with the quartic circulant graphs and disclose several results regarding their null spaces. First of all, we derive an explicit formula for computing the nullity of any quartic circulant graph. Furthermore, we provide the full nut graph characterization among these graphs. Finally, we determine all such graphs that attain the minimum or maximum nullity with respect to a given order. We also give the extremal null spaces of these graphs.