Nut graphs with a prescribed number of vertex and edge orbits
Nino Bašić, Ivan Damnjanović
TL;DR
The paper resolves the realizability problem for nut graphs with prescribed vertex and edge orbits. It first constructs infinite families of Cayley nut graphs with $o_e(G)=o_a(G)=k$ for any $k\ge2$, and then extends these results to realize all pairs with $o_v(G)=r$ and $o_e(G)=k$ for even $r$ and all $k\ge r+1$ (with odd $r$ handled via subdivision from the Cayley cases). A subdivision lemma shows how to augment $o_v$ while controlling $o_e$ and $o_a$, enabling a complete Buset-type realization for $(o_v,o_e)$ and establishing infinite realizability. The work also derives corollaries for Cayley nut graphs and identifies open problems, notably the full characterization of pairs $(o_v,o_a)$. The results deepen understanding of the interplay between graph automorphisms, orbit structures, and spectral properties in nut graphs, with constructive methods based on circulant and Cayley graphs.
Abstract
A nut graph is a nontrivial graph whose adjacency matrix has a one-dimensional null space spanned by a vector without zero entries. Recently, it was shown that a nut graph has more edge orbits than vertex orbits. It was also shown that for any even $r \ge 2$ and any $k \ge r + 1$, there exist infinitely many nut graphs with $r$ vertex orbits and $k$ edge orbits. Here, we extend this result by finding all the pairs $(r, k)$ for which there exists a nut graph with $r$ vertex orbits and $k$ edge orbits. In particular, we show that for any $k \ge 2$, there are infinitely many Cayley nut graphs with $k$ edge orbits and $k$ arc orbits.