The Laplacian of a uniform hypergraph
Abstract In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a $$k$$-uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval $$[0,2]$$, and the real part is zero (respectively two) if and only if the eigenvalue is...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Hu, Shenglong [verfasserIn] |
|---|
| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2013 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media New York 2013 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of combinatorial optimization - Springer US, 1997, 29(2013), 2 vom: 05. Feb., Seite 331-366 |
|---|---|
| Übergeordnetes Werk: |
volume:29 ; year:2013 ; number:2 ; day:05 ; month:02 ; pages:331-366 |
| Links: |
|---|
| DOI / URN: |
10.1007/s10878-013-9596-x |
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OLC2044618281 |
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| 520 | |a Abstract In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a $$k$$-uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval $$[0,2]$$, and the real part is zero (respectively two) if and only if the eigenvalue is zero (respectively two). All the H$$^+$$-eigenvalues of the Laplacian and all the smallest H$$^+$$-eigenvalues of its sub-tensors are characterized through the spectral radii of some nonnegative tensors. All the H$$^+$$-eigenvalues of the Laplacian that are less than one are completely characterized by the spectral components of the hypergraph and vice verse. The smallest H-eigenvalue, which is also an H$$^+$$-eigenvalue, of the Laplacian is zero. When $$k$$ is even, necessary and sufficient conditions for the largest H-eigenvalue of the Laplacian being two are given. If $$k$$ is odd, then its largest H-eigenvalue is always strictly less than two. The largest H$$^+$$-eigenvalue of the Laplacian for a hypergraph having at least one edge is one; and its nonnegative eigenvectors are in one to one correspondence with the flower hearts of the hypergraph. The second smallest H$$^+$$-eigenvalue of the Laplacian is positive if and only if the hypergraph is connected. The number of connected components of a hypergraph is determined by the H$$^+$$-geometric multiplicity of the zero H$$^+$$-eigenvalue of the Lapalacian. | ||
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10.1007/s10878-013-9596-x doi (DE-627)OLC2044618281 (DE-He213)s10878-013-9596-x-p DE-627 ger DE-627 rakwb eng 510 VZ 3,2 ssgn Hu, Shenglong verfasserin aut The Laplacian of a uniform hypergraph 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2013 Abstract In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a $$k$$-uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval $$[0,2]$$, and the real part is zero (respectively two) if and only if the eigenvalue is zero (respectively two). All the H$$^+$$-eigenvalues of the Laplacian and all the smallest H$$^+$$-eigenvalues of its sub-tensors are characterized through the spectral radii of some nonnegative tensors. All the H$$^+$$-eigenvalues of the Laplacian that are less than one are completely characterized by the spectral components of the hypergraph and vice verse. The smallest H-eigenvalue, which is also an H$$^+$$-eigenvalue, of the Laplacian is zero. When $$k$$ is even, necessary and sufficient conditions for the largest H-eigenvalue of the Laplacian being two are given. If $$k$$ is odd, then its largest H-eigenvalue is always strictly less than two. The largest H$$^+$$-eigenvalue of the Laplacian for a hypergraph having at least one edge is one; and its nonnegative eigenvectors are in one to one correspondence with the flower hearts of the hypergraph. The second smallest H$$^+$$-eigenvalue of the Laplacian is positive if and only if the hypergraph is connected. The number of connected components of a hypergraph is determined by the H$$^+$$-geometric multiplicity of the zero H$$^+$$-eigenvalue of the Lapalacian. Tensor Eigenvalue Hypergraph Laplacian Qi, Liqun aut Enthalten in Journal of combinatorial optimization Springer US, 1997 29(2013), 2 vom: 05. Feb., Seite 331-366 (DE-627)216539323 (DE-600)1339574-9 (DE-576)094421935 1382-6905 nnns volume:29 year:2013 number:2 day:05 month:02 pages:331-366 https://doi.org/10.1007/s10878-013-9596-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_24 GBV_ILN_26 GBV_ILN_70 GBV_ILN_2108 AR 29 2013 2 05 02 331-366 |
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10.1007/s10878-013-9596-x doi (DE-627)OLC2044618281 (DE-He213)s10878-013-9596-x-p DE-627 ger DE-627 rakwb eng 510 VZ 3,2 ssgn Hu, Shenglong verfasserin aut The Laplacian of a uniform hypergraph 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2013 Abstract In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a $$k$$-uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval $$[0,2]$$, and the real part is zero (respectively two) if and only if the eigenvalue is zero (respectively two). All the H$$^+$$-eigenvalues of the Laplacian and all the smallest H$$^+$$-eigenvalues of its sub-tensors are characterized through the spectral radii of some nonnegative tensors. All the H$$^+$$-eigenvalues of the Laplacian that are less than one are completely characterized by the spectral components of the hypergraph and vice verse. The smallest H-eigenvalue, which is also an H$$^+$$-eigenvalue, of the Laplacian is zero. When $$k$$ is even, necessary and sufficient conditions for the largest H-eigenvalue of the Laplacian being two are given. If $$k$$ is odd, then its largest H-eigenvalue is always strictly less than two. The largest H$$^+$$-eigenvalue of the Laplacian for a hypergraph having at least one edge is one; and its nonnegative eigenvectors are in one to one correspondence with the flower hearts of the hypergraph. The second smallest H$$^+$$-eigenvalue of the Laplacian is positive if and only if the hypergraph is connected. The number of connected components of a hypergraph is determined by the H$$^+$$-geometric multiplicity of the zero H$$^+$$-eigenvalue of the Lapalacian. Tensor Eigenvalue Hypergraph Laplacian Qi, Liqun aut Enthalten in Journal of combinatorial optimization Springer US, 1997 29(2013), 2 vom: 05. Feb., Seite 331-366 (DE-627)216539323 (DE-600)1339574-9 (DE-576)094421935 1382-6905 nnns volume:29 year:2013 number:2 day:05 month:02 pages:331-366 https://doi.org/10.1007/s10878-013-9596-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_24 GBV_ILN_26 GBV_ILN_70 GBV_ILN_2108 AR 29 2013 2 05 02 331-366 |
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10.1007/s10878-013-9596-x doi (DE-627)OLC2044618281 (DE-He213)s10878-013-9596-x-p DE-627 ger DE-627 rakwb eng 510 VZ 3,2 ssgn Hu, Shenglong verfasserin aut The Laplacian of a uniform hypergraph 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2013 Abstract In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a $$k$$-uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval $$[0,2]$$, and the real part is zero (respectively two) if and only if the eigenvalue is zero (respectively two). All the H$$^+$$-eigenvalues of the Laplacian and all the smallest H$$^+$$-eigenvalues of its sub-tensors are characterized through the spectral radii of some nonnegative tensors. All the H$$^+$$-eigenvalues of the Laplacian that are less than one are completely characterized by the spectral components of the hypergraph and vice verse. The smallest H-eigenvalue, which is also an H$$^+$$-eigenvalue, of the Laplacian is zero. When $$k$$ is even, necessary and sufficient conditions for the largest H-eigenvalue of the Laplacian being two are given. If $$k$$ is odd, then its largest H-eigenvalue is always strictly less than two. The largest H$$^+$$-eigenvalue of the Laplacian for a hypergraph having at least one edge is one; and its nonnegative eigenvectors are in one to one correspondence with the flower hearts of the hypergraph. The second smallest H$$^+$$-eigenvalue of the Laplacian is positive if and only if the hypergraph is connected. The number of connected components of a hypergraph is determined by the H$$^+$$-geometric multiplicity of the zero H$$^+$$-eigenvalue of the Lapalacian. Tensor Eigenvalue Hypergraph Laplacian Qi, Liqun aut Enthalten in Journal of combinatorial optimization Springer US, 1997 29(2013), 2 vom: 05. Feb., Seite 331-366 (DE-627)216539323 (DE-600)1339574-9 (DE-576)094421935 1382-6905 nnns volume:29 year:2013 number:2 day:05 month:02 pages:331-366 https://doi.org/10.1007/s10878-013-9596-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_24 GBV_ILN_26 GBV_ILN_70 GBV_ILN_2108 AR 29 2013 2 05 02 331-366 |
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10.1007/s10878-013-9596-x doi (DE-627)OLC2044618281 (DE-He213)s10878-013-9596-x-p DE-627 ger DE-627 rakwb eng 510 VZ 3,2 ssgn Hu, Shenglong verfasserin aut The Laplacian of a uniform hypergraph 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2013 Abstract In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a $$k$$-uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval $$[0,2]$$, and the real part is zero (respectively two) if and only if the eigenvalue is zero (respectively two). All the H$$^+$$-eigenvalues of the Laplacian and all the smallest H$$^+$$-eigenvalues of its sub-tensors are characterized through the spectral radii of some nonnegative tensors. All the H$$^+$$-eigenvalues of the Laplacian that are less than one are completely characterized by the spectral components of the hypergraph and vice verse. The smallest H-eigenvalue, which is also an H$$^+$$-eigenvalue, of the Laplacian is zero. When $$k$$ is even, necessary and sufficient conditions for the largest H-eigenvalue of the Laplacian being two are given. If $$k$$ is odd, then its largest H-eigenvalue is always strictly less than two. The largest H$$^+$$-eigenvalue of the Laplacian for a hypergraph having at least one edge is one; and its nonnegative eigenvectors are in one to one correspondence with the flower hearts of the hypergraph. The second smallest H$$^+$$-eigenvalue of the Laplacian is positive if and only if the hypergraph is connected. The number of connected components of a hypergraph is determined by the H$$^+$$-geometric multiplicity of the zero H$$^+$$-eigenvalue of the Lapalacian. Tensor Eigenvalue Hypergraph Laplacian Qi, Liqun aut Enthalten in Journal of combinatorial optimization Springer US, 1997 29(2013), 2 vom: 05. Feb., Seite 331-366 (DE-627)216539323 (DE-600)1339574-9 (DE-576)094421935 1382-6905 nnns volume:29 year:2013 number:2 day:05 month:02 pages:331-366 https://doi.org/10.1007/s10878-013-9596-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-WIW SSG-OPC-MAT GBV_ILN_24 GBV_ILN_26 GBV_ILN_70 GBV_ILN_2108 AR 29 2013 2 05 02 331-366 |
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the laplacian of a uniform hypergraph |
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The Laplacian of a uniform hypergraph |
| abstract |
Abstract In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a $$k$$-uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval $$[0,2]$$, and the real part is zero (respectively two) if and only if the eigenvalue is zero (respectively two). All the H$$^+$$-eigenvalues of the Laplacian and all the smallest H$$^+$$-eigenvalues of its sub-tensors are characterized through the spectral radii of some nonnegative tensors. All the H$$^+$$-eigenvalues of the Laplacian that are less than one are completely characterized by the spectral components of the hypergraph and vice verse. The smallest H-eigenvalue, which is also an H$$^+$$-eigenvalue, of the Laplacian is zero. When $$k$$ is even, necessary and sufficient conditions for the largest H-eigenvalue of the Laplacian being two are given. If $$k$$ is odd, then its largest H-eigenvalue is always strictly less than two. The largest H$$^+$$-eigenvalue of the Laplacian for a hypergraph having at least one edge is one; and its nonnegative eigenvectors are in one to one correspondence with the flower hearts of the hypergraph. The second smallest H$$^+$$-eigenvalue of the Laplacian is positive if and only if the hypergraph is connected. The number of connected components of a hypergraph is determined by the H$$^+$$-geometric multiplicity of the zero H$$^+$$-eigenvalue of the Lapalacian. © Springer Science+Business Media New York 2013 |
| abstractGer |
Abstract In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a $$k$$-uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval $$[0,2]$$, and the real part is zero (respectively two) if and only if the eigenvalue is zero (respectively two). All the H$$^+$$-eigenvalues of the Laplacian and all the smallest H$$^+$$-eigenvalues of its sub-tensors are characterized through the spectral radii of some nonnegative tensors. All the H$$^+$$-eigenvalues of the Laplacian that are less than one are completely characterized by the spectral components of the hypergraph and vice verse. The smallest H-eigenvalue, which is also an H$$^+$$-eigenvalue, of the Laplacian is zero. When $$k$$ is even, necessary and sufficient conditions for the largest H-eigenvalue of the Laplacian being two are given. If $$k$$ is odd, then its largest H-eigenvalue is always strictly less than two. The largest H$$^+$$-eigenvalue of the Laplacian for a hypergraph having at least one edge is one; and its nonnegative eigenvectors are in one to one correspondence with the flower hearts of the hypergraph. The second smallest H$$^+$$-eigenvalue of the Laplacian is positive if and only if the hypergraph is connected. The number of connected components of a hypergraph is determined by the H$$^+$$-geometric multiplicity of the zero H$$^+$$-eigenvalue of the Lapalacian. © Springer Science+Business Media New York 2013 |
| abstract_unstemmed |
Abstract In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a $$k$$-uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval $$[0,2]$$, and the real part is zero (respectively two) if and only if the eigenvalue is zero (respectively two). All the H$$^+$$-eigenvalues of the Laplacian and all the smallest H$$^+$$-eigenvalues of its sub-tensors are characterized through the spectral radii of some nonnegative tensors. All the H$$^+$$-eigenvalues of the Laplacian that are less than one are completely characterized by the spectral components of the hypergraph and vice verse. The smallest H-eigenvalue, which is also an H$$^+$$-eigenvalue, of the Laplacian is zero. When $$k$$ is even, necessary and sufficient conditions for the largest H-eigenvalue of the Laplacian being two are given. If $$k$$ is odd, then its largest H-eigenvalue is always strictly less than two. The largest H$$^+$$-eigenvalue of the Laplacian for a hypergraph having at least one edge is one; and its nonnegative eigenvectors are in one to one correspondence with the flower hearts of the hypergraph. The second smallest H$$^+$$-eigenvalue of the Laplacian is positive if and only if the hypergraph is connected. The number of connected components of a hypergraph is determined by the H$$^+$$-geometric multiplicity of the zero H$$^+$$-eigenvalue of the Lapalacian. © Springer Science+Business Media New York 2013 |
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