  Let $H$ be a connected bipartite graph, whose signless Laplacian matrix is
$Q(H)$. Suppose that the bipartition of $H$ is $(S,T)$ and that $x$ is the
eigenvector of the smallest eigenvalue of $Q(H)$. 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.
