Universality of the fully connected vertex in Laplacian continuous-time quantum walk problems

Abstract A fully connected vertex w in a simple graph G of order N is to all the other ? 1 vertices. Upon denoting by L Laplacian matrix graph, we prove that continuous-time quantum walk (CTQW)—with Hamiltonian H = ?L —of walker initially localized at | ? does not depend on . We also for any Grover-like CTQW—with + ? ? ?? |—the probability amplitude marked vertices The result hold CTQW with ?A (adjacency matrix). apply our results spatial search and transport single multiple vertices, proving CTQWs inherit properties already known complete same order, including optimality search. Our provide unified framework several partial reported literature such as equivalence central star wheel graph.