On the sign patterns of the smallest signless Laplacian eigenvector

Let H   be a connected bipartite graph, whose signless Laplacian matrix is Q(H)Q(H). Suppose that the bipartition of H   is (S,T)(S,T) and that x   is the eigenvector of the smallest eigenvalue of Q(H)Q(H). It is well-known that x is positive and constant on S, and negative and constant on T.The resilience of the sign pattern of x under addition of edges into the subgraph induced by either S or T is investigated and a number of cases in which the sign pattern of x persists are described.