Abstract
We generalize Laplacian matrices for graphs to Laplacian tensors for even uniform hypergraphs and set some foundations for the spectral hypergraph theory based upon Laplacian tensors. Especially, algebraic connectivity of an even uniform hypergraph based on Z-eigenvalues of the corresponding Laplacian tensor is introduced and its connections with edge connectivity and vertex connectivity are discussed.
Similar content being viewed by others
References
Berge C (1973) Hypergraphs. Combinatorics of finite sets, 3rd edn. North-Holland, Amsterdam
Cartwright D, Sturmfels B (2011) The number of eigenvalues of a tensor. To appear in: Linear Algebra Appl
Chung FRK (1997) Spectral graph theory. Am. Math. Soc., Providence
Fiedler M (1973) Algebraic connectivity of graphs. Czech Math J 23(98):298–305
Horn R, Johnson CR (1985) Matrix analysis. Cambridge University Press, New York
Lim L-H (2005) Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE international workshop on computational advances in multi-sensor adaptive processing, CAMSAP ’05, 2005, vol 1, pp 129–132
Lim L-H (2007) Foundations of numerical multilinear algebra: decomposition and approximation of tensors. PhD thesis, Standford University, USA
Lim L-H (2011) Eigenvalues and eigenvectors of Cholesky decomposable tensors. Talk on JRI workshop on eigenvalues of nonnegative tensors, December 18, 2011. The Hong Kong Polytechnic University
Merris R (1994) Laplacian matrics of graphs: a survey. Linear Algebra Appl 198:143–176
Nemhauser GL, Wolsey LA (1988) Integer programming and combinatorial optimization. Wiley, New York
Ni G, Qi L, Wang F, Wang Y (2007) The degree of the e-characteristic polynomial of an even order tensor. J Math Anal Appl 329:1218–1229
Qi L (2005) Eigenvalues of a real supersymmetric tensor. J Symb Comput 40:1302–1324
Qi L (2007) Eigenvalues and invariants of tensors. J Math Anal Appl 325:1363–1377
Reznick B (1992) Sums of even powers of real linear forms. Mem. AMS 96(463)
Rota Bulò S (2009) A game-theoretic framework for similarity-based data clustering. PhD thesis, Università Ca’ Foscari di Venezia, Italy
Rota Bulò S, Pelillo M (2009) A generalization of the Motzkin-Straus theorem to hypergraphs. Optim Lett 3:187–295
Author information
Authors and Affiliations
Corresponding author
Additional information
L. Qi was supported by the Hong Kong Research Grant Council.
Rights and permissions
About this article
Cite this article
Hu, S., Qi, L. Algebraic connectivity of an even uniform hypergraph. J Comb Optim 24, 564–579 (2012). https://doi.org/10.1007/s10878-011-9407-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-011-9407-1