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 relation1-ρ 1 2=<minB RBorel-setP D (X B)=<1/2X,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.