The zero eigenvalue of the Laplacian tensor of a uniform hypergraph

Abstract

In this paper, we study that the algebraic multiplicity of the zero Laplacian eigenvalue of a connected uniform hypergraph. We give the algebraic multiplicity of the zero Laplacian eigenvalue of a hyperstar. For a loose hyperpath, we characterize the algebraic multiplicity of the zero Laplacian eigenvalue by the multiplicities of points in the affine variety defined by the Laplacian eigenvalue equations. We compute the algebraic multiplicities of the zero Laplacian eigenvalue of a loose hyperpath. We also show that the algebraic multiplicity of the zero Laplacian eigenvalue is not smaller than the number of irreducible components of the eigenvariety associated with the zero Laplacian eigenvalue for a loose hyperpath.

Keywords:

• hypergraph
• Laplacian tensor
• resultant
• algebraic multiplicity

AMS Classifications:

• 05C65
• 15A18
• 15A69

Acknowledgments

The author would like to thank the editor and anonymous referee for their valuable comments and suggestions, which have considerably improved the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).