Signless-Laplacian-PST for vertex-complemented coronas of K2 at arbitrary orders

Determine whether the vertex-complemented corona K₂◦H avoids signless-Laplacian perfect state transfer between the two vertices of K₂ for every order k of a regular graph H.

Background

The paper studies signless-Laplacian perfect state transfer (PST) on vertex-complemented coronae. For a regular graph H of order k, the authors prove that K₂◦H does not admit signless-Laplacian-PST when k=1 or when k is prime. They also invoke an earlier result establishing the same nonexistence for even k. These cases motivate the unresolved question of whether the nonexistence result holds for every positive integer k, including composite odd orders that are not covered by the stated results.