Abstract
For a graph G of order n, let λ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that λ2(G)+λ2(G‾)≥1, where G‾ is the complement of G. For any x∈Rn, let ∇x∈R(n2) be the vector whose {i,j}-th entry is |xi−xj|. In this paper, we show the aforementioned conjecture is equivalent to prove that every two orthonormal vectors f,g∈Rn with zero mean satisfy‖∇f−∇g‖2≥2. In this article, it is shown that for the validity of the conjecture it suffices to prove that the conjecture holds for all permutation graphs.
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