A note on a distance bound using eigenvalues of the normalized Laplacian matrix (original) (raw)

Let G be a connected graph, and let X and Y be subsets of its vertex set. A previously published bound is considered that relates the distance between X and Y to the eigenvalues of the normalized Laplacian matrix for G, the volumes of X and Y , and the volumes of their complements. A counterexample is given to the bound, and then a corrected version of the bound is provided. http://math.technion.ac.il/iic/ela ELA A Distance Bound 205 the context). The normalized Laplacian for H(p, q) is given by   I −1 √ q(p+q−1) J −1 √ q(p+q−1) J p+q p+q−1 I − 1 p+q−1 J