Polyenes with maximum HOMO–LUMO gap

On the basis of a variable neighbourhood search with the AutoGraphiX software, it is conjectured that for even numbers of atoms the fully conjugated acyclic π system of maximum HOMO–LUMO gap is a `combʹ in which each vertex of a backbone carries a single pendant edge. Chemically, this represents CnH3n/2+2, an α,ω-diene with methylene groups attached at all intermediate positions. As n increases without limit, the conjectured asymptotic dimensionless gap is 22−2. Odd-numbered graphs that would have maximum gap as singly charged π systems are reported for up to 41 vertices; they are related to comb graphs.