Some Spectral Properties of Uniform Hypergraphs

  • Jiang Zhou
  • Lizhu Sun
  • Wenzhe Wang
  • Changjiang Bu
Keywords: Hypergraph eigenvalue, Adjacency tensor, Laplacian tensor, Signless Laplacian tensor, Power hypergraph


For a $k$-uniform hypergraph $H$, we obtain some trace formulas for the Laplacian tensor of $H$, which imply that $\sum_{i=1}^nd_i^s$ ($s=1,\ldots,k$) is determined by the Laplacian spectrum of $H$, where $d_1,\ldots,d_n$ is the degree sequence of $H$. Using trace formulas for the Laplacian tensor, we obtain expressions for some coefficients of the Laplacian polynomial of a regular hypergraph. We give some spectral characterizations of odd-bipartite hypergraphs, and give a partial answer to a question posed by Shao et al (2014). We also give some spectral properties of power hypergraphs, and show that a conjecture posed by Hu et al (2013) holds under certain conditons.
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