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We are interested in the clustering problem on graphs: it is known that if there are two underlying clusters, then the signs of the eigenvector corresponding to the second largest eigenvalue of the adjacency matrix can reliably reconstruct the two clusters.
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K. Burdzy, The hot spots problem in planar domains with one hole, Duke Math. J. 129 (2005), p. 481–502
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Moo Chung, Seongho Seo, Nagesh Adluru, and Houri Vorperian. Hot spots conjecture and its application to modeling tubular structures. In Kenji Suzuki, Fei Wang, Dinggang Shen, and Pingkun Yan, editors, Machine Learning in Medical Imaging, volume 7009 of Lecture Notes in Computer Science, p. 225–232
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C. Judge and S. Mondal, Euclidean Triangles Have No Hot Spots, Ann. of Math, to appear
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