Spectrum of the 1‐Laplacian and Cheeger's Constant on Graphs |
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Authors: | K C Chang |
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Institution: | LMAM SCHOOL OF MATHEMATICAL SCIENCES, PEKING UNIVERSITY, BEIJING, CHINA |
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Abstract: | We develop a nonlinear spectral graph theory, in which the Laplace operator is replaced by the 1 ? Laplacian Δ1. The eigenvalue problem is to solve a nonlinear system involving a set valued function. In the study, we investigate the structure of the solutions, the minimax characterization of eigenvalues, the multiplicity theorem, etc. The eigenvalues as well as the eigenvectors are computed for several elementary graphs. The graphic feature of eigenvalues are also studied. In particular, Cheeger's constant, which has only some upper and lower bounds in linear spectral theory, equals to the first nonzero Δ1 eigenvalue for connected graphs. |
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Keywords: | spectral graph theory 1‐Laplace operator Cheeger's constant critical point theory 05C50 05C35 58E05 58C40 05C75 |
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