On 12-regular nut graphs Article Swipe

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nut combinatorics circulant matrix mathematics adjacency matrix graph discrete mathematics eigenvalues and eigenvectors strongly regular graph regular graph line graph graph power physics quantum mechanics acoustics

Nino Bašić , Martin Knor , Riste Škrekovski ·

YOU? · · 2021 · Open Access · · DOI: https://doi.org/10.26493/2590-9770.1403.1b1 · OA: W3126445281

A nut graph is a simple graph whose adjacency matrix is singular with $1$-dimensional kernel such that the corresponding eigenvector has no zero entries. In 2020, Fowler et al. characterised for each $d \in \{3,4,\ldots,11\}$ all values $n$ such that there exists a $d$-regular nut graph of order $n$. In the present paper, we determine all values $n$ for which a $12$-regular nut graph of order $n$ exists. We also present a result by which there are infinitely many circulant nut graphs of degree $d \equiv 0 \pmod 4$ and no circulant nut graph of degree $d \equiv 2 \pmod 4$.