On cubic polycirculant nut graphs

Abstract A nut graph is a nontrivial simple graph whose adjacency matrix contains a one-dimensional null space spanned by a vector without zero entries. Moreover, an $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> -circulant graph is a graph that admits a cyclic group of automorphisms having $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> vertex orbits of equal size. It is not difficult to observe that there exists no cubic 1-circulant nut graph or cubic 2-circulant nut graph, while the full classification of all the cubic 3-circulant nut graphs was recently obtained (Damnjanović et al. in Electron J Comb 31(2):P2.31, 2024). Here, we investigate the existence of cubic $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> -circulant nut graphs for $$\ell \ge 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> and show that there is no cubic 4-circulant nut graph or cubic 5-circulant nut graph by using a computer-assisted proof. Furthermore, we rely on a construction based approach in order to demonstrate that there exist infinitely many cubic $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> -circulant nut graphs for any fixed $$\ell \in \{6, 7 \}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>{</mml:mo> <mml:mn>6</mml:mn> <mml:mo>,</mml:mo> <mml:mn>7</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> or $$\ell \ge 9$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>9</mml:mn> </mml:mrow> </mml:math> .