Sharp lower bounds on the spectral radius of uniform hypergraphs concerning degrees

Let $\mathcal{A}(H)$ and $\mathcal{Q}(H)$ be the adjacency tensor and signless Laplacian tensor of an $r$-uniform hypergraph $H$. Denote by $\rho(H)$ and $\rho(\mathcal{Q}(H))$ the spectral radii of $\mathcal{A}(H)$ and $\mathcal{Q}(H)$, respectively. In this paper we present a lower bound on $\rho(H)$ in terms of vertex degrees and we characterize the extremal hypergraphs attaining the bound, which solves a problem posed by Nikiforov [V. Nikiforov, Analytic methods for uniform hypergraphs, Linear Algebra Appl. 457 (2014) 455-535]. Also, we prove a lower bound on $\rho(\mathcal{Q}(H))$ concerning degrees and give a characterization of the extremal hypergraphs attaining the bound.


Introduction
Let G = (V (G), E(G)) be a simple undirected graph with n vertices, and A(G) be the adjacency matrix of G. Let ρ(G) be the spectral radius of G, and d i be the degree of vertex i of G, i = 1, 2, . . ., n. In 1988, Hofmeister [6] obtained a lower bound on ρ(G) in terms of degrees of vertices of G as follows: Furthermore, if G is connected, then equality holds if and only if G is either a regular graph or a semiregular bipartite graph (see details in [6] and [19]). The inequality (1) has many important applications in spectral graph theory (see [4,8,9]).
In recent years the research on spectra of hypergraphs via tensors have drawn increasingly extensive interest, accompanying with the rapid development of tensor spectral theory. A hypergraph H = (V, E) consists of a (finite) set V and a collection E of non-empty subsets of V (see [1]). The elements of V are called vertices and the elements of E are called hyperedges, a path if all vertices and edges in the walk are distinct. A hypergraph H is called connected if for any vertices i, j, there is a walk connecting i and j. For positive integers r and n, a real tensor A = (a i 1 i 2 ···ir ) of order r and dimension n refers to a multidimensional array (also called hypermatrix) with entries a i 1 i 2 ···ir such that a i 1 i 2 ···ir ∈ R for all i 1 , i 2 , . . ., i r ∈ [n]. We say that tensor A is symmetric if its entries a i 1 i 2 ···ir are invariant under any permutation of its indices.
Recently, Nikiforov [10] presented some analytic methods for studying uniform hypergraphs, and posed the following question (see [10,Question 11.5]): , is the degree of vertex i, and ρ(Q(H)) is the spectral radius of the signless Laplacian tensor Q(H). Then with equality if and only if H is regular.

Preliminaries
In this section we review some basic notations and necessary conclusions. Denote the set of nonnegative vectors (positive vectors) of dimension n by R n + (R n ++ ). The unit tensor of order r and dimension n is the tensor I n = (δ i 1 i 2 ···ir ), whose entry is 1 if i 1 = i 2 = · · · = i r and 0 otherwise.
From the above definition, let x = (x 1 , x 2 , . . . , x n ) T be a column vector of dimension n. Then Ax is a vector in C n , whose i-th component is as the following In 2005, Lim [7] and Qi [12] independently introduced the concepts of tensor eigenvalues and the spectra of tensors. Let A be an order r and dimension n tensor, x = (x 1 , x 2 . . . , x n ) T ∈ C n be a column vector of dimension n. If there exists a number λ ∈ C and a nonzero vector x ∈ C n such that then λ is called an eigenvalue of A, x is called an eigenvector of A corresponding to the The spectral radius ρ(A) of A is the maximum modulus of the eigenvalues of A. It was proved that λ is an eigenvalue of A if and only if it is a root of the characteristic polynomial of A (see details in [16]).
In 2012, Cooper and Dutle [3] defined the adjacency tensors for r-uniform hypergraphs. 3,14]). Let H = (V (H), E(H)) be an r-uniform hypergraph on n vertices. The adjacency tensor of H is defined as the order r and dimension n tensor A(H) = (a i 1 i 2 ···ir ), Let D(H) be an order r and dimension n diagonal tensor with its diagonal element d ii···i being For an r-uniform hypergraph H, denote the spectral radius of A(H) by ρ(H). It should be announced that spectral radius defined in [10] differ from this paper, while for an r-uniform hypergraph H the spectral radius defined in [10] equals to (r − 1)!ρ(H). This is not essential and does not effect the result.
In [5], the weak irreducibility of nonnegative tensors was defined. It was proved that an runiform hypergraph H is connected if and only if its adjacency tensor A(H) is weakly irreducible (see [5] and [18]). Clearly, this shows that if H is connected, then A(H), L(H) and Q(H) are all weakly irreducible. The following result for nonnegative tensors is stated as a part of Perron-Frobenius theorem in [2].
). Let A be a nonnegative tensor of order r and dimension n. Then we have the following statements.
(1) ρ(A) is an eigenvalue of A with a nonnegative eigenvector corresponding to it.
(2) If A is weakly irreducible, then ρ(A) is the unique eigenvalue of A with the unique eigenvector x ∈ R n ++ , up to a positive scaling coefficient. 13]). Let A be a nonnegative symmetric tensor of order r and dimension n. Then we have Furthermore, x ∈ R n + with ||x|| r = 1 is an optimal solution of the above optimization problem if and only if it is an eigenvector of A corresponding to the eigenvalue ρ(A).
The following concept of direct products (also called Kronecker product) of tensors was defined in [15], which is a generalization of the direct products of matrices.

Definition 2.3 ([15]
). Let A and B be two order r tensors with dimension n and m, respectively.
Define the direct product A ⊗ B to be the following tensor of order r and dimension mn (the set of subscripts is taken as [n] × [m] in the lexicographic order): In particular, if x = (x 1 , x 2 , . . . , x n ) T and y = (y 1 , y 2 , . . . , y m ) T are two column vectors with dimension n and m, respectively. Then The following basic results can be found in [15].    Proof of Claim 1. It suffices to show that for any (i, j), (s, t) ∈ V ( H), there exists a walk connecting them. We distinguish the following two cases.
Since H is connected, there exists a path i = i 1 e 1 i 2 · · · i ℓ e ℓ i ℓ+1 = s. Since r 3, there exist If ℓ is even, we obtain Hence there exists a walk connecting (i, j) and (s, t).
Since r 3, there exist i ′ and j ′ such that i ′ = i, j ′ = j, j ′ = t. According to Case 1 we know that there is a path connecting (i, j) and (i ′ , j ′ ). Noting that i ′ = s and j ′ = t, there is a path connecting (i ′ , j ′ ) and (s, t) by Case 1. So there exists a walk connecting (i, j) and (s, t), as desired. The proof of the claim is completed.

Claim 2. A( H) = A(H) ⊗ B.
Proof of Claim 2. From the definition of H, it follows that According to Definition 2.3, A(H) ⊗ B is an order r and dimension rn tensor, whose entries are  On the other hand, let z = (z 1 , z 2 , . . . , z r ) ∈ R r + be a nonnegative eigenvector corresponding to ρ(B) with ||z|| r = 1. By AM-GM inequality, we have with equality holds if and only if Therefore, ρ(B) = (r − 1)!. The proof of the claim is completed.
Notice that the maximal absolute value of the roots of a complex polynomial is a continuous function on the coefficients of the polynomial. Take the limit ε → 0 on both sides of the above equation, we obtain the desired result. The proof of the claim is completed.
It is clear that H is an r-partite hypergraph with partition We define a vector x ∈ R rn as follows: where a 1 , a 2 , . . ., a n 0 and a r 1 + a r 2 + · · · + a r n = n. It is obvious that d H ((i, j)) = (r − 1)!d i for any i ∈ [n], j ∈ [r]. By Theorem 2.2, we deduce that j 1 ),...,(ir,jr)}∈E( H) It follows from a r 1 + a r 2 + · · · + a r n = n and Hölder inequality that with equality if and only if Now we set a i as (5). In the light of (4) and (5) we have Now we give a characterization of extremal hypergraphs achieving the equality in (6). Suppose first the equality holds in (6). Then the vector x ∈ R rn defined by (3)  Consequently, u 1 = u 2 = · · · = u n , which implies that H is regular.
Conversely, if H is a connected regular hypergraph, it is easy to see that the equality (6) holds.
Remark 3.1. It is known that the spectral radius ρ(H) of H is greater than or equal to the average degree [3], i.e., It follows from PM inequality that Therefore Theorem 1.1 has a better estimation for spectral radius of H.

Proof of Theorem 1.2
In this section, we shall give a proof of Theorem 1.2.
Proof of Theorem 1.2. Let H be a connected r-uniform hypergraph with vertex set V (H) = [n]. Let H be the r-uniform hypergraph as defined in Theorem 1.1. Suppose that B is the order r and dimension r tensor given by (2), and I r is the unit tensor of order r and dimension r.
We have the following claims.
If i 1 = i 2 = · · · = i r = i, i ∈ [n] and j 1 = j 2 = · · · = j r = j, then The proof of the claim is completed. Let x ∈ R rn be the column vector defined by (3). By Theorem 2.2, we have Furthermore, by AM-GM inequality, we have a r i + r − 1 = a r i + 1 + 1 + · · · + 1 r−1 So it follows from (4), (7) and (8) that By (5) Suppose that the equality holds in (9). Then the vector x ∈ R rn defined by (3) is an eigenvector corresponding to ρ(Q( H)) by Theorem 2.2 and a i = 1, i ∈ [n] by (8). Recall that Q( H) is weakly irreducible. By Claim 6, u ⊗ e is a positive eigenvector to ρ(Q( H)). We see that x and u ⊗ e are linear dependence by Theorem 2.1. Therefore, d 1 = d 2 = · · · = d n , which implies that H is regular.
It was proved that if H is a connected r-uniform hypergraph, then ρ(L(H)) = ρ(Q(H)) if and only if r is even and H is odd-colorable [17]. Therefore we have ρ(L(H)) 2 1 n n i=1 d r r−1 i r−1 r for a connected odd-colorable hypergraph H, which generalizes the result in [19].