Laplacian state transfer in graphs with involutions

For 𝑞 ∈ ℝ ﹨ { 0 } , the generalized Laplacian of a graph X is the matrix ℒ = Δ + 𝑞 ⁢ 𝐴 , where Δ is the degree matrix and A is the adjacency matrix of X . This paper investigates perfect state transfer (PST) on graphs with possible loops equipped with non-trivial involutions, where the generalized Laplacian matrix is taken as the Hamiltonian of the underlying spin network. We establish an equivalence between the existence of PST between pair states (or plus states) in such a graph and PST between vertices in a subgraph induced by the involution. This allows us to prove that almost all simple unweighted planar graphs (resp., almost all simple unweighted trees) can be assigned loops of weight one to exactly two vertices to produce PST between pair states relative to ℒ . We also construct an infinite family of graphs with loops admitting PST between pair or plus states at arbitrarily long distances. For paths on n vertices, we show that PST occurs between end vertices relative to ℒ if and only if 𝑛 = 2 , or ( 𝑛 , 𝑞 ) = ( 3 , √ 𝑘 2 − ℓ 2 8 ⁢ ℓ 2 ) , where k and ℓ are integers with 𝑘 > ℓ and 𝑘 ≢ ℓ ⁡ ( m o d ⁡ 2 ) . For cycles, we show that the addition of an edge does not yield vertex PST relative to Laplacian and signless Laplacian matrices. Finally, we show that the addition of a few suitable edges (including loops) in complete bipartite graphs, cycles, and paths yields PST between pair states.