H-Eigenvalues of signless Laplacian tensor for an even uniform hypergraph

doi:10.1007/s11464-012-0266-6

Front Math Chin

0, Vol.

Issue () : 107-127 https://doi.org/10.1007/s11464-012-0266-6

RESEARCH ARTICLE

H-Eigenvalues of signless Laplacian tensor for an even uniform hypergraph

Jinshan XIE^1,2, An CHANG¹(

)

1. Center for Discrete Mathematics, Fuzhou University, Fuzhou 350003, China; 2. School of Mathematics and Computer Science, Longyan University, Longyan 364012, China

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Abstract

The signless Laplacian tensor and its H-eigenvalues for an even uniform hypergraph are introduced in this paper. Some fundamental properties of them for an even uniform hypergraph are obtained. In particular, the smallest and the largest H-eigenvalues of the signless Laplacian tensor for an even uniform hypergraph are discussed, and their relationships to hypergraph bipartition, minimum degree, and maximum degree are described. As an application, the bounds of the edge cut and the edge connectivity of the hypergraph involving the smallest and the largest H-eigenvalues are presented.

Keywords Signless Laplacian tensor hypergraph H-eigenvalue bipartition maximum degree bound edge cut

Corresponding Author(s): CHANG An,Email:anchang@fzu.edu.cn

Issue Date: 01 February 2013

Cite this article:

Jinshan XIE,An CHANG. H-Eigenvalues of signless Laplacian tensor for an even uniform hypergraph[J]. Front Math Chin, 0, (): 107-127.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-012-0266-6
https://academic.hep.com.cn/fmc/EN/Y0/V/I/107

1	Brouwer A E, Haemers W H. Spectra of Graphs. Berlin: Springer, 2011
2	Cartwright D, Sturmfels B. The number of eigenvalues of a tensor. Linear Algebra Appl , 2013, 438(2): 942-952 doi: 10.1016/j.laa.2011.05.040
3	Chang K C, Pearson K, Zhang T. Perron Frobenius Theorem for nonnegative tensors. Commun Math Sci , 2008, 6: 507-520
4	Chang K C, Pearson K, Zhang T. Primitivity, the convergence of the NQZ method, and the largest eigenvalue for nonnegative tensors. School of Mathematical Sciences , Peking University, Beijing, China, Preprint, 2010
5	Chang K C, Pearson K, Zhang T. Some variational principles of the Z-eigenvalues for nonnegative tensors. School of Mathematical Sciences , Peking University, Beijing, China, Preprint, 2011. http://www.mathinst.pku.edu.cn/data/upload/month 201112/ 2011-no44 dp47rB.pdf
6	Chang K C, Qi L, Zhou G. Singular values of a real rectangular tensor. J Math Anal Appl , 2010, 370: 284-294 doi: 10.1016/j.jmaa.2010.04.037
7	Chung F R K. Spectral Graph Theory. Providence: Amer Math Soc, 1997
8	Cooper J, Dutle A. Spectra of uniform hypergraphs. Linear Algebra Appl , 2012, 436(9): 3268-3292 doi: 10.1016/j.laa.2011.11.018
9	Cvetkovi? D, Rowlinson P, Simi? S K. Eigenvalue bounds for the signless Laplacian. Publ Inst Math (Beograd) , 2007, 95(81): 11-27 doi: 10.2298/PIM0795011C
10	Fiedler M. Algebraic connectivity of graphs. Czech Math J , 1973, 98(23): 298-305
11	Hu S, Qi L. Algebraic connectivity of an even uniform hypergraph. J Comb Optim , 2012, 24(4): 564-579 doi: 10.1007/s10878-011-9407-1
12	Kolda T G, Bader B W. Tensor decompositions and applications. SIAM Review , 2009, 51: 455-500 doi: 10.1137/07070111X
13	Lim L-H. Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE InternationalWorkshop on Computational Advances in Multi- Sensor Adaptive Processing (CAMSAP ’05) , Vol 1. 2005, 129-132
14	Ni G, Qi L, Wang F, Wang Y. The degree of the E-characteristic polynomial of an even order tensor. J Math Anal Appl , 2007, 329: 1218-1229 doi: 10.1016/j.jmaa.2006.07.064
15	Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput , 2005, 40: 1302-1324 doi: 10.1016/j.jsc.2005.05.007
16	Qi L. Eigenvalues and invariants of tensors. J Math Anal Appl , 2007, 325: 1363-1377 doi: 10.1016/j.jmaa.2006.02.071
17	Qi L. The spectral theory of tensors. arXiv: 1201.3424v1 [math.SP]
18	Qi L, Sun W, Wang Y. Numerical multilinear algebra and its applications. Front Math China , 2007, 2(4): 501-526 doi: 10.1007/s11464-007-0031-4
19	Rota Bulò S, Pelillo M. A generalization of the Motzkin-Straus theorem to hypergraphs. Optim Lett , 2009, 3: 287-295 doi: 10.1007/s11590-008-0108-3
20	Rota Bulò S, Pelillo M. New bounds on the clique number of graphs based on spectral hypergraph theory. In: Stützle T, ed. Learning and Intelligent Optimization. Berlin: Springer-Verlag, 2009, 45-48 doi: 10.1007/978-3-642-11169-3_4
21	Van Loan C. Future directions in tensor-based computation and modeling. Workshop Report in Arlington, Virginia at National Science Foundation , February20-21, 2009. http://www.cs.cornell.edu/cv/TenWork/Home.htm
22	Yang Q, Yang Y. Further results for Perron-Frobenius Theorem for nonnegative tensors II. SIAM J Matrix Anal Appl , 2011, 32: 1236-1250 doi: 10.1137/100813671
23	Yang Y, Yang Q. Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J Matrix Anal Appl , 2010, 31: 2517-2530 doi: 10.1137/090778766
24	Yang Y, Yang Q. Singular values of nonnegative rectangular tensors. Front Math China , 2011, 6(2): 363-378 doi: 10.1007/s11464-011-0108-y

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