On the sign patterns of the smallest signless Laplacian eigenvector

Abstract

Let H be a connected bipartite graph, whose signless Laplacian matrix is Q(H).

Perturbations of discrete elliptic operators

“…In [7,Definition 1.3], given H ⊂ V a maximal independent set, they say the network Γ is H-Roth if any non null eigenfunction u ∈ C(V ) corresponding to λ, the lowest eigenvalue of Q, satisfies that u > 0 on H and u < 0 on V \ H, or vice versa. Newly, this property of the eigenfunctions implies that λ is simple and hence that Q is elliptic, since it is positive-semidefinite.…”

“…Whereas Roth's theorem is a generalization to some Schrödinger-like operators, of the behavior of the signless Laplacian on bipartite networks, Goldberg and Kirkland show that Roth's theorem can be generalized to the signless Laplacian on networks obtained from a bipartite network by adding edges whose ends are in the same bipartite component. Specifically, [7,Corollary 6.5] says that if K s,t is the complete bipartite graph and s ≥ t, then the graph Γ obtained from K s,t by adding edges between vertices in H, the bipartite component of cardinality s, is H-Roth. Some results concerning to the case s < t are also studied in [7].…”

“…Specifically, [7,Corollary 6.5] says that if K s,t is the complete bipartite graph and s ≥ t, then the graph Γ obtained from K s,t by adding edges between vertices in H, the bipartite component of cardinality s, is H-Roth. Some results concerning to the case s < t are also studied in [7].…”

“…In this work we are only interested in Schrödinger-like operators whose principal part is a singular Laplacian-like operator. The remaining Schrödinger-like operators, including those studied in [7] will be treated in future works.…”