Work

On the degrees of regular nut graphs and Cayley nut graphs

Year: 2026

Type: article

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 ≤ d ≤ 12 and for d ≥ 4 such that d ≡ 0 (mod 4). Here it is shown that infinitely many d-regular nut graphs exist for each degree d ≥ 3. Moreover, we prove that there are infinitely many d-regular Cayley nut graphs for each even d ≥ 4. This implies that we have identified all feasible degrees d for which a d-regular Cayley nut graph exists.

Source: Ars Mathematica Contemporanea

DOI: 10.26493/1855-3974.3641.49c

Authors: Nino Bašić, Ivan Damnjanović, Patrick W. Fowler

Institutions: University of Primorska, Institute of Mathematics, Physics, and Mechanics, University of Nis, University of Sheffield

Language: English

Topic: Graph theory and applications

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: Leverhulme Trust, European Commission, Javna Agencija za Raziskovalno Dejavnost RS, Science Fund of the Republic of Serbia

Awards: Higher-level bibliographic services, Computationally intensive methods in theoretical computer science, discrete mathematics, combinatorial optimization, and numerical analysis and algebra with applications in natural and social sciences, Mathematical and computational methods for polyhedral self-assembly