Some spectral properties of uniform hypergraphs

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 \cite{ShaoShanWu}. We also give some spectral properties of power hypergraphs, and show that a conjecture posed by Hu et al \cite{HuQiShao} holds under certain conditons.

Let A be an order k 2 dimension n tensor, and let x = (x 1 , . . . , x n ) ⊤ . From Definition 1.1, the product Ax is a vector in C n whose i-th component is (see Example 1.1 in [15]) The concept of tensor eigenvalues was posed in [9,12]. If there exists a nonzero vector x ∈ C n such that Ax = λx [k−1] , then λ is called an eigenvalue of A, x is an eigenvector of λ, where x [k−1] = (x k−1 1 , . . . , x k−1 n ) ⊤ . The determinant of A, denoted by det(A), is the resultant of the system of polynomials f i (x 1 , . . . , x n ) = (Ax) i (i = 1, . . . , n). The characteristic polynomial of A is defined as φ A (λ) = det(λI n − A), where I n is the unit tensor of order k and dimension n. It is known that eigenvalues of A are exactly roots of φ A (λ) [12]. The multiset of roots of φ A (λ) (counting multiplicities) is the spectrum of A, denoted by σ(A). The maximal modulus of eigenvalues of A is called the spectral radius of A, denoted by ρ(A). More details on eigenvalues and characteristic polynomials of tensors can be found in [4,12].
A hypergraph H is called k-uniform if each edge of H contains exactly k distinct vertices. Let V (H) and E(H) denote the vertex set and the edge set of H, respectively. In [13], Qi defined the Laplacian and the signless Laplacian tensor of a uniform hypergraph as follows.
Definition 1.2. [7,13] The adjacency tensor of a k-uniform hypergraph H, denoted by A H , is an order k dimension |V (H)| tensor with entries This paper is organized as follows. In Section 2, we give some trace formulas for the Laplacian tensor of a uniform hypergraph, and obtain expressions for some coefficients of the Laplacian polynomial of a regular hypergraph. In Section 3, we give some spectral characterizations of odd-bipartite hypergraphs. In Section 4, we give some spectral properties of power hypergraphs.

Laplacian spectra and degree sequence of hypergraphs
Traces of tensors are very useful in the study of spectral theory of tensors. The d-th order trace of an order k 2 dimension n tensor T = (t i 1 ···i k ) is defined as [1,4,10] T r d (T ) = (k − 1) n−1 where A = (a ij ) is an n × n auxiliary matrix, and ∂ ∂a iy = ∂ ∂a ii 2 · · · ∂ ∂a ii k if y = i 2 · · · i k . The codegree d coefficient of the characteristic polynomial of T can be expressed in terms of T r 1 (T ), . . . , T r d (T ) (see [4,Theorem 6.3]). It is also known that T r t (T ) = λ∈σ(T ) λ t for any t ∈ [n(k − 1) n−1 ] (see [4,Theorem 6.10]). Hence T r d (T ) is an important invariant in the spectral theory of tensors.
Shao et al [16] give a graph theoretical formula for T r d (T ). In order to describe this formula, we introduce some notations in [16]. For an integer d > 0, we define For F = (i 1 α 1 , . . . , i d α d ) ∈ F d and an order k 2 dimension n tensor T = (t i 1 ···i k ), we write π F (T ) = d j=1 t i j α j . Let p i (F ) be the total number of times that the index i appears in F . If p i (F ) is a multiple of k for any i ∈ [n], then F is called k-valent. Shao et al give a graph theoretical formula for T r d (T ) as follows (see equation (3.5) in [16]).
For a k-uniform hypergraph H, Cooper and Dutle [1] proved that T r d (A H ) = 0 for d ∈ [k − 1]. We give some trace formulas for the Laplacian (signless Laplacian) tensor of uniform hypergraphs as follows.
Proof. By Lemma 2.2, we have where (1) and Definition 2.1, we have Similar with the above procedure, we can also get (1) and Definition 2.1, we have Similar with the above procedure, we can also get Remark. Note that traces of a tensor are determined by its spectrum [ Let p t (M) denote the codegree t coefficient of the characteristic polynomial of a tensor M.
Proof. From [4, Theorem 6.10], we have We can obtain the expression of t!p t (M) from Cramer's rule.
A uniform hypergraph H is called d-regular if each vertex of H has degree d. The following are some coefficients of the Laplacian (signless Laplacian) polynomial of regular hypergraphs.
Theorem 2.5. Let H be a d-regular k-uniform hypergraph with n vertices. Then where T r t = T r t (L H ). Since H is d-regular, by Theorem 2.3, we have T r i = dT r i−1 = n(k − 1) n−1 d i , i = 2, . . . , k − 1. If t < k, then by Eq. (2), we have Since T r 1 = n(k − 1) n−1 d, we have Similar with the above procedure, we can also get Similar with the above procedure, we can also get

Eigenvalues and odd-bipartite hypergraphs
A k-uniform hypergraph H is called odd-bipartite, if there exists a proper subset V 1 of V (H) such that each edge of H contains exactly odd number of vertices in V 1 [6,17]. Spectral characterizations of odd-bipartite hypergraphs will be investigated in this section. We first give some auxiliary lemmas. The following lemma can be obtained from equation (2.1) in [15].
Lemma 3.1. Let A = (a i 1 ···i k ) be an order k 2 dimension n tensor, and let P = (p ij ), Q = (q ij ) be n × n matrices. Then

Lemma 3.2. [6] Let H be a connected k-uniform hypergraph. A nonzero vector x is an eigenvector of Q H corresponds to the zero eigenvalue if and only if there exist nonzero γ ∈ C and integers
for some integer σ e associated with each e ∈ E(H).
Weakly irreducible tensors are defined in [3]. It is known that a k-uniform hypergraph H is connected if and only if A H is weakly irreducible [11]. Lemma 3.3. [17,19] Let A be an order k dimension n nonnegative weakly irreducible tensor. If ρ(A) exp(θ √ −1) is an eigenvalue of A, then there exists a diagonal matrix U with unit diagonal entries such that For a tensor T , let Hσ(T ) = {λ|λ ∈ σ(T ), λ has a real eigenvector}. For a connected k-uniform hypergraph G, Shao et al [17] proved that Shao et al wish to know whether the reverse implication is true. We show that the reverse is true when k is not divisible by 4. Proof. From [17, Theorem 2.2], we have (1) ⇒ (2) ⇒ (3). Since 0 is always an eigenvalue of L G (see [13]), we have (3) ⇒ (4). Next we prove that (4) ⇒ (1).
If 0 is an eigenvalue of Q G , then by Lemma 3.2, there exists a vertex labeling f : for each e ∈ E(G). Hence k is even. Since k is not divisible by 4, we know that k 2 is odd. So i∈e f (i) is odd for each e ∈ E(G).
For any e ∈ E(G), since i∈e f (i) is odd, e contains exactly odd number of vertices in V 1 . Hence G is odd-bipartite.
When k = 2, Theorem 3.4 becomes a classic result in spectral graph theory, i.e., a connected graph G is bipartite if and only if 0 is a signless Laplacian eigenvalue of G. It is also well known that a connected graph G is bipartite if and only if −ρ(A G ) is an eigenvalue of G. We generalize this result as follows.
where u i j is the diagonal entry of U corresponds to vertex i j (j = 1, . . . , k). For any edge e = i 1 i 2 · · · i k ∈ E(H), we get Similarly, we have u i 1 u i 2 · · · u i k = −u k i 1 = · · · = −u k i k . Since u i 1 , . . . , u i k are unit complex number, there exist integers α i 1 , . . . , α i k and θ such that Then Hence k j=1 α i j ≡ k 2 (mod k), k is even. Since k is not divisible by 4, k j=1 α i j is odd for any edge e = i 1 i 2 · · · i k ∈ E(H). Let V 1 = {u|u ∈ V (H), α u is odd}. For any e = i 1 i 2 · · · i k ∈ E(H), since k j=1 α i j is odd, e contains exactly odd number of vertices in V 1 . Hence H is odd-bipartite.
Let H be a connected k-uniform hypergraph. If 0 is an eigenvalue of Q H , then by the proof of Theorem 3.4, we know that there exists a vertex labeling for each e ∈ E(H). We pose the following conjecture.

Eigenvalues of power hypergraphs
A vertex with degree one is called a core vertex [7]. For a k-uniform hypergraph H, if e ∈ E(H) contains core vertices, then we use H − e to denote a k-uniform sub-hypergraph of H obtained by deleting the edge e and all core vertices in e.
Some examples of power hypergraphs are given in [7, Fig.1]. From Definition 4.2, we know that each edge of a power hypergraph G k contains two adjacent vertices in V (G) and k − 2 core vertices not in V (G).
If H is a connected k-uniform hypergraph, then A H and Q H are both weakly irreducible [13]. So we obtain the following lemma from [13, Theorem 2.2].  Proof. Suppose that x is an eigenvector of the eigenvalue λ = 0 of graph G. Then j∈N G (i) x j = λx i for any i ∈ V (G), where N G (i) is the set of all neighbors of i in G. Let y be a column vector of dimension |V (G k )| such that is a core vertex in the edge contains two adjacent vertices i, j ∈ V (G). For any Hence λ 2 k is an eigenvalue of G k with an eigenvector y.
If G is connected and λ = ρ(A G ), then we can choose x as a positive eigenvector of ρ(A G ). In this case, y is a positive eigenvector of the eigenvalue ρ(A G ) 2 k of G k . Lemma 4.3 implies that ρ(A G k ) = ρ(A G ) 2 k when G is connected.
If G has r 2 components G 1 , . . . , G r , then ρ(A G k ) = max{ρ(A G k 1 ), . . . , ρ(A G k r )} = max{ρ(A G 1 ) We can obtain the following result from Theorem 4.4.  Let P n and S n be the path and the star of order n, respectively. The following result was proved by Li et al [8]. Here we give a different proof. Proof. Among all trees with n vertices, P n is the unique tree with the smallest adjacency spectral radius, and S n is the unique tree with the largest adjacency spectral radius [2]. By Theorem 4.4, we have