## Optimization problems for weighted graphs and related correlation estimates.(English)Zbl 1047.05026

Authors’ abstract: $$L_2$$ and $$L_1$$-norm optimization problems for weighted graphs are discussed and compared in the paper. In the setting where the standardized weight matrix $$W$$ is regarded as a joint probability distribution $$D$$ of two discrete random variables with equal marginals, the refined upper bound of $$\sqrt{\lambda_1(2-\lambda_1)}$$ for the Cheeger constant gives the relation $\frac{1-\rho_1}2\leq \min_{B\subset{\mathbb R}} P_W(X'\in\overline B| X\in B)\leq \sqrt{1-\rho_1^2},$ where $$B$$ is a Borel set, $$P_D(X\in B)\leq 1/2$$, $$X,X'$$ i.d., with the symmetric maximal correlation $$\rho_1$$, provided that it is positive, or equivalently, for the smallest positive eigenvalue of the weighted Laplacian $$\lambda_1\leq 1$$ holds.

### MSC:

 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 60C05 Combinatorial probability 62H20 Measures of association (correlation, canonical correlation, etc.)
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### References:

 [1] Alon, N.; Milman, V.D., λ1, isoperimetric inequalities for graphs and superconcentrators, J. combin. theory B, 38, 73-88, (1985) · Zbl 0549.05051 [2] Bolla, M., Singular values decomposition of linear operators on Hilbert-spaces, Alkalmazott mat. lapok, 13, 189-206, (1987 88), (in Hungarian) [3] Bolla, M., Correspondence analysis, Alkalmazott mat. lapok, 13, 207-230, (1987 88), (in Hungarian) [4] Bolla, M., Spectra, Euclidean representations and clusterings of hypergraphs, Discrete math., 117, 19-39, (1993) · Zbl 0781.05036 [5] Bolla, M.; Tusnády, G., Spectra and optimal partitions of weighted graphs, Discrete math., 128, 1-20, (1994) · Zbl 0796.05066 [6] Breiman, L.; Friedman, H.J., Estimating optimal transformations for multiple regression and correlation, J. amer. statist. assoc., 80, 580-619, (1985) · Zbl 0594.62044 [7] Buser, P., On the bipartition of graphs, Discrete appl. math., 9, 105-109, (1984) · Zbl 0544.05038 [8] Cheeger, J., A lower bound for the smallest eigenvalue of the Laplacian, (), 195-199 · Zbl 0212.44903 [9] Chung, F.R.K., Spectral graph theory, CBMS regional conference series in mathematics, Vol. 92, (1994), AMS Publications Providence, RI · Zbl 0872.05052 [10] Diaconis, P.; Stroock, D.W., Geometric bounds for eigenvalues of Markov chains, Ann. appl. probab., 1, 36-61, (1991) · Zbl 0731.60061 [11] A. Lubotzky, Discrete groups, expanding graphs, and invariant measures, Appendix by J.D. Rogawski, Progress in Mathematics, Vol. 125, Birkhäuser, Basel, Boston, Berlin, 1994. · Zbl 0826.22012 [12] Mohar, B., Isoperimetric numbers of graphs, J. combin. theory B, 47, 274-291, (1989) · Zbl 0719.05042 [13] Papadimitriou, C.H.; Raghavan, P.; Tamaki, H.; Vempala, S., Latent semantic indexinga probabilistic analysis, J. comput. system sci., 61, 217-235, (2000) · Zbl 0963.68063 [14] Rényi, A., On measures of dependence, Acta math. acad. sci. hung., 10, 441-451, (1959) · Zbl 0091.14403 [15] Sinclair, A.; Jerrum, M., Approximate counting, uniform generation and rapidly mixing Markov chains, Inform. and comput., 82, 93-133, (1989) · Zbl 0668.05060
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