Optimization problems for weighted graphs and related correlation estimates

Abstract

L 2- and L 1-norm optimization problems for weighted graphs are discussed and compared in the paper. The standardized weight matrix W is also regarded as a joint probability distribution of two discrete random variables with equal marginals, D. In this setting, the refined upper bound λ 1(2−λ 1) for the Cheeger constant gives the relation 1−ρ 1 2 ⩽ min B⊂ R Borel-set P D(X∈B)⩽1/2 X,X′ i.d. P W(X′∈ B ̄ |X∈B)⩽ 1−ρ 1 2 with the symmetric maximal correlation ρ 1, provided that it is positive, or equivalently, for the smallest positive eigenvalue of the weighted Laplacian λ 1⩽1 holds.