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# Communications in Mathematical Sciences

## Volume 12 (2014)

### Number 6

### H^{+}-eigenvalues of Laplacian and signless Laplacian tensors

Pages: 1045 – 1064

DOI: http://dx.doi.org/10.4310/CMS.2014.v12.n6.a3

#### Author

#### Abstract

We propose a simple and natural definition for the Laplacian and the signless Laplacian tensors of a uniform hypergraph. We study their $\mathrm{H}^{+}$-eigenvalues, i.e., $\mathrm{H}$-eigenvalues with nonnegative $\mathrm{H}$-eigenvectors, and $\mathrm{H}^{++}$-eigenvalues, i.e., $\mathrm{H}$-eigenvalues with positive $\mathrm{H}$-eigenvectors. We show that each of the Laplacian tensor, the signless Laplacian tensor, and the adjacency tensor has at most one $\mathrm{H}^{++}$-eigenvalue, but has several other $\mathrm{H}^{+}$-eigenvalues. We identify their largest and smallest $\mathrm{H}^{+}$-eigenvalues, and establish some maximum and minimum properties of these $\mathrm{H}^{+}$-eigenvalues. We then define analytic connectivity of a uniform hypergraph and discuss its application in edge connectivity.

#### Keywords

Laplacian tensor, signless Laplacian tensor, uniform hypergraph, $\mathrm{H}^{+}$-eigenvalue

#### 2010 Mathematics Subject Classification

05C65, 15A18