On nut graphs with two vertex and three edge orbits

TL;DR

This paper introduces a new construction method for nut graphs with specific symmetry properties, confirming a conjecture for many odd non-prime orders and providing numerous examples and structural insights.

Contribution

The paper presents a general construction for nut graphs with two vertex and three edge orbits, verifying the conjecture for most odd non-prime orders up to a million.

Findings

Confirmed the conjecture for all odd non-prime orders up to 2,500.

Verified the conjecture for at least 99.8% of odd non-prime orders up to one million.

Provided structural and spectral conditions for nut graphs with desired symmetry properties.

Abstract

Nut graphs are graphs whose adjacency matrix is singular with one-dimensional null space spanned by a vector with no zero entries. In a recent paper, Ba\v{s}i\'c, Fowler and Pisanski proved that the automorphism group of a nut graph has more orbits on the edge set than on the vertex set. They classified all orders for which a vertex-transitive nut graph with precisely two edge orbits exists, and conjectured that a nut graph with two vertex and three edge orbits exists for each non-prime order $n \ge 9$. Motivated by this conjecture, we introduce a very general construction that provides graphs with the desired symmetry properties, and we determine some sufficient spectral and structural conditions under which they are nut graphs. The construction yields infinite families of examples and confirms the above conjecture for all odd non-prime orders up to $2\,500$ and for at least $99.8$ percent of all odd non-prime orders up to a million. Finally, we present some additional interesting examples of nut graphs with two vertex and three edge orbits that do not arise from this construction.