Abstract
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.
Highlights
The research on spectral theory of hypergraphs has attracted extensive attention [1,5,6,7,8,11,13,14,16,17,18]
An order k dimension n tensor A = ∈ Cn×···×n is a multidimensional array with nk entries, where ij ∈ [n], j = 1, . . . , k
The characteristic polynomial of A is defined as φA(λ) = det(λIn − A), where In is the unit tensor of order k and dimension n
Summary
The research on spectral theory of hypergraphs has attracted extensive attention [1,5,6,7,8,11,13,14,16,17,18]. The product AB is the following tensor C of order (m − 1)(k − 1) + 1 and dimension n with entries c = iα1...αm−1 a b ii2...im i2α1 · · · bimαm−1 The characteristic polynomial of A is defined as φA(λ) = det(λIn − A), where In is the unit tensor of order k and dimension n. In [13], Qi defined the Laplacian and the signless Laplacian tensor of a uniform hypergraph as follows. [7, 13] The adjacency tensor of a k-uniform hypergraph H, denoted by AH, is an order k dimension |V (H)| tensor with entries ai1i2···ik =. Let DH be an order k dimension |V (H)| diagonal tensor whose diagonal entries are vertex degrees of H.
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