Spectral theory of Laplace operators on oriented hypergraphs
Several new spectral properties of the normalized Laplacian defined for oriented hypergraphs are shown. The eigenvalue 1 and the case of duplicate vertices are discussed; two Courant nodal domain theorems are established; new quantities that bound the eigenvalues are introduced. In particular, the C...
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| Published in | Discrete mathematics Vol. 344; no. 6; p. 112372 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
01.06.2021
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| Abstract | Several new spectral properties of the normalized Laplacian defined for oriented hypergraphs are shown. The eigenvalue 1 and the case of duplicate vertices are discussed; two Courant nodal domain theorems are established; new quantities that bound the eigenvalues are introduced. In particular, the Cheeger constant is generalized and it is shown that the classical Cheeger bounds can be generalized for some classes of hypergraphs; it is shown that a geometric quantity used to study zonotopes bounds the largest eigenvalue from below, and that the notion of coloring number can be generalized and used for proving a Hoffman-like bound. Finally, the spectrum of the unnormalized Laplacian for Cartesian products of hypergraphs is discussed. |
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| AbstractList | Several new spectral properties of the normalized Laplacian defined for oriented hypergraphs are shown. The eigenvalue 1 and the case of duplicate vertices are discussed; two Courant nodal domain theorems are established; new quantities that bound the eigenvalues are introduced. In particular, the Cheeger constant is generalized and it is shown that the classical Cheeger bounds can be generalized for some classes of hypergraphs; it is shown that a geometric quantity used to study zonotopes bounds the largest eigenvalue from below, and that the notion of coloring number can be generalized and used for proving a Hoffman-like bound. Finally, the spectrum of the unnormalized Laplacian for Cartesian products of hypergraphs is discussed. |
| ArticleNumber | 112372 |
| Author | Mulas, Raffaella Zhang, Dong |
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| Cites_doi | 10.13001/1081-3810.1625 10.1016/0026-2692(92)90064-8 10.1016/j.laa.2008.01.029 10.1016/j.jctb.2005.10.002 10.1016/S0024-3795(01)00313-5 10.1016/j.laa.2019.05.008 10.1134/S0001434621010120 10.1016/j.aim.2019.05.025 10.37236/2763 10.1007/s101070100233 10.1016/j.laa.2012.06.011 10.1016/j.laa.2018.07.012 10.1103/PhysRevE.101.062313 |
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| Keywords | Laplace operator Hoffman bound Oriented hypergraphs Spectral theory Cheeger inequality Chromatic number |
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| SubjectTerms | Cheeger inequality Chromatic number Hoffman bound Laplace operator Oriented hypergraphs Spectral theory |
| Title | Spectral theory of Laplace operators on oriented hypergraphs |
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