Spectral extremal results for hypergraphs

Let F be a graph. A hypergraph is called Berge F if it can be obtained by replacing each edge in F by a hyperedge containing it. Given a family of graphs F , we say that a hypergraph H is Berge F -free if for every F ∈ F , the hypergraph H does not contain a Berge F as a subhypergraph. In this paper we investigate the connections between spectral radius of the adjacency tensor and structural properties of a linear hypergraph. In particular, we obtain a spectral version of Tur´an-type problems over linear k -uniform hypergraphs by using spectral methods, including a tight result on Berge C 4 -free linear 3-uniform hypergraphs.


Introduction
A hypergraph H = (V, E) consists of a vertex set V and an edge (hyperedge) set E, where each edge is a nonempty subset of V . A hypergraph is called k-uniform if each edge is a k-element subset of V . A 2-uniform hypergraph is simply called a graph. Two vertices x and y are said to be adjacent if there is an edge that contains both of these vertices. A hypergraph H is called linear if every two edges have at most one vertex in common.
For a fixed k-uniform family F, the Turán number of F, denoted by ex k (n, F), is the maximum number of edges of an F-free hypergraph on n vertices. Similarly, given a family of k-uniform linear hypergraphs F, the linear Turán number of F, denoted ex lin k (n, F), is the maximum number of edges in an F-free k-uniform linear hypergraph on n vertices.
Turán type extremal problems in graphs and hypergraphs are the central topic of extremal combinatorics and have a vast literature. For a survey of recent results we refer the reader to [9,16,20]. Only a handful of results are known about the asymptotic behaviour of Turán numbers for hypergraphs. One of the most active subjects is Berge hypergraphs. The classical definition of a hypergraph cycle due to Berge is the following: a Berge cycle C t of length t ≥ 2 is an alternating sequence of distinct vertices (other than first and last) and distinct edges of the form v 1 l 1 v 2 l 2 · · · v t l t where v i , v i+1 ∈ l i for each i ∈ {1, 2, · · · , t − 1} and v t , v 1 ∈ l t . Gerbner and Palmer [12] gave the following natural generalization of the definitions of Berge graphs. Let F = (V (F ), E(F )) be a graph and B = (V (B), E(B)) be a hypergraph. We say B is Berge F if there is a bijection φ : E(F ) → E(B) such that e ⊆ φ(e) for all e ∈ E(F ). In other words, given a graph F , we can obtain a Berge F by replacing each edge of F with a hyperedge that contains it. Given a family of graphs F, we say that a hypergraph H is Berge F-free if for every F ∈ F, the hypergraph H does not contain a Berge F as a subhypergraph. The maximum possible number of edges in a Berge F-free hypergraph on n vertices is the Turán number of Berge F.
Our aim is to consider a spectral version of hypergraph Turán problems, i.e., spectral extremal hypergraph theory, which is the subset of extremal problems where invariants are based on the eigenvalues or eigenvectors of a hypergraph. The most natural such invariant is the maximal absolute value of the eigenvalues of the adjacency tensor of a hypergraph H, called its spectral radius. Because the spectral radius is a close correlate of the number of edges in a hypergraph, the following problem is a natural spectral analog of hypergraph Turán problems: What is the maximum spectral radius of hypergraphs of order n, not containing a given F?
In fact, spectral extremal graph theory has a substantial history, with many important results. Examples include Stanley's bound [24], theorems of Wilf [26] relating eigenvalues of graphs to their chromatic number, and many other examples. Much of recent work in spectral extremal graph theory is due to Nikiforov, who has considered maximizing the spectral radius over several families of graphs; see [21]. Although the area is still difficult and underdeveloped, we believe that ultimately a spectral approach to extremal hypergraph theory will turn out to be a fruitful and interesting accompaniment to "conventional" extremal theory, see e.g., [1,15].
The rest of this paper is organized as follows. In the next section, the bulk of the necessary notation and the basic facts have been presented, including the definitions and properties of eigenvalues of tensors and hypergraphs. We develop a few new tools in order to provide spectral analogues of extremal hypergraph problems. In Section 3 we consider spectral analogues of Turán-type problems for hypergraphs and investigate bounds on the maximum spectral radius of linear k-uniform hypergraphs with girth at least five by using the spectral methods proved in Section 2.

Spectra of tensors
In 2005, Qi [22] and Lim [18] independently introduced the concept of tensor eigenvalues and the spectra of tensors. An order k dimension n real tensor 1 T = (T i 1 ···i k ) consists of n k real entries T i 1 ···i k for 1 ≤ i 1 , i 2 , · · · , i k ≤ n. Evidently, a vector of dimension n is a tensor of order 1 and a matrix is a tensor of order 2. T is called symmetric if the value of T i 1 ···i k is invariant under any permutation of the indices i 1 , i 2 , · · · , i k . Given a vector x ∈ R n , T x k is a real number and T x k−1 is an n-dimensional vector defined as follows. T x k and the ith component of T x k−1 are given by: (1) The former is simply tensor contraction of T with the k-th outer product of x with itself, and the latter is the i-th coordinate of the contraction of T with the (k − 1)-st outer product of x with itself. Note that the symmetry of T makes these contractions well-defined without specifying which indices are summed over. Let T be an order k dimension n real tensor. For some λ ∈ C, if there exists a nonzero vector x ∈ C n satisfying the eigenequation then λ is a called an eigenvalue of T and x is its corresponding eigenvector, where . If x is a real eigenvector of T , then clearly the corresponding eigenvalue λ is real. In this case, λ is called an H-eigenvalue and x is called an Heigenvector associated with λ. Furthermore, if x is nonnegative and real, we say λ is an H + -eigenvalue of T . If x is positive and real, λ is said to be an H ++ -eigenvalue of T . The maximal absolute value of the eigenvalues of T is called the spectral radius of T , denoted by ρ(T ).
In 2012, Cooper and Dutle [5] defined the adjacency tensor of a k-uniform hypergraph H. The adjacency tensor A = A(H) is an order k dimension n 1 Sometimes known as a "hypermatrix" or simply, "matrix." symmetric tensor defined by otherwise.
This definition generalizes adjacency matrices, and its theory is a natural starting point for spectral hypergraph theory. For a vector x of dimension n and a subset U ⊆ V , we write The product Ax k therefore has an interpretation as follows: The right-hand side (without the factor of k) is sometimes known as the Lagrangian polynomial of H.
For nonnegative tensors, we have a generalization of the Perron-Frobenius theorem, see [3,5,11,27]. Let T = (T i 1 ···i k ) be an order k dimension n nonnegative tensor. If for any nonempty proper index subset α ⊂ {1, 2, · · · , n}, there is at least an entry T i 1 ···i k > 0, where i 1 ∈ α and at least an i j / ∈ α for j = 2, · · · , k, then T is called nonnegative weakly irreducible tensor. It was proved that a k-uniform hypergraph H is connected if and only if its adjacency tensor A(H) is weakly irreducible (see [11,27]). Let ρ(H) denote the spectral radius of a hypergraph H. By the Perron-Frobenius theorem, if H is connected, the eigenvector x = (x 1 , x 2 , · · · , x n ) T corresponding to ρ(H), known as the principal eigenvector, can be chosen to be strictly positive. Throughout the paper, we only consider connected and simple hypergraphs.

Useful tools
In this section we present some useful tools which help to recast into spectral theory some classical results on hypergraphs and their proofs. We also introduce some additional notation employed below.
Let H = (V, E) be a connected simple hypergraph on n vertices and m edges. For a vertex v, let L(v) be the set of edges containing v and N v be the neighborhood of v, i.e., The degree of a vertex v, which is denoted by d(v), is defined as the number of edges containing v, i.e., d(v) = |L(v)|. For any two vertices u and v, let N uv be the set of common neighbors of u and v. The codegree of u and v, denoted by d(u, v), is the number of edges containing both u and v in H. Let ∆, ∆ 2 and d denote the maximum degree, the maximum codegree and the average degree of H, respectively. It is easy to verify that d = km/n.
If X and Y are disjoint sets of vertices of H, we write E(X, Y ) as the set of edges containing some vertices in X and the others in Y and e(X, Y ) the number of edges in Lemma 2.1. Let H be a connected simple k-uniform hypergraph and ρ be the spectral radius of the adjacency tensor of H. Then where u is the vertex corresponding to a maximum entry of the principal eigenvector.
Proof. Let x be an eigenvector corresponding to ρ. For a vertex v ∈ V (H), we will use x v to denote the eigenvector entry of x corresponding to v. For any v ∈ V (H), the eigenvector equation is By the Perron-Frobenius Theorem, x has all positive entries, and it will be convenient for us to normalize so that the maximum entry of x is 1. Throughout the paper, we will use u to denote the vertex with maximum eigenvector entry equal to 1. If there are multiple such vertices, choose and fix u arbitrarily among them. Since x u = 1, (6) becomes Apply AM-GM inequality to (7), we have The next inequality is a simple consequence of our normalization and an easy double counting argument, but will be used extensively throughout the paper and therefore warrants special attention. Multiplying both sides of (8) by ρ and applying (6) gives Since the maximum entry of x is normalized to 1, it is obvious that x e\{v} ≤ 1 for any vertex v ∈ V (H). Thus We now estimate the right side of (9) in two different ways. On one hand, for a fixed edge {u, i 2 , · · · , i k } ∈ E, we consider the edge set L(i j ) for 2 ≤ j ≤ k. The edge set L(i j ) can be considered the union of the k disjoint subsets of edges incident to i j which have intersection with N u of cardinality t = 1 through t = k. That is, (9), this gives On the other hand, for an arbitrary edge l = {i 1 , i 2 , · · · , i k } with |l ∩ N u | = t, without loss of generality, assume l ∩ N u = {i 1 , i 2 , · · · , i t }. Then the contribution of the edge l on the right-hand side of (10) is just the sum of the number of edges containing both the vertices u and i j over 1 ≤ j ≤ t. In other words, the edge l appears in the right summation exactly d(u, i 1 ) + d(u, i 2 ) + · · · + d(u, i t ) times.
Thus (10) becomes This completes the proof.
Note that Lemma 2.1 illustrates a relationship between spectral radius of the adjacency tensor and structural properties of hypergraphs. Combined with estimates of the codegree, one can obtain results such as the following.
Lemma 2.2. Let H be a connected simple k-uniform hypergraph with maximum codegree ∆ 2 and ρ be the spectral radius of the adjacency tensor of H. Let u be the vertex with maximum eigenvector entry. Then Proof. Since the maximum codegree of H is ∆ 2 , by Lemma 2.1, we have where the last equality follows because both sums count the number of pairs It is clear that the codegree of each pair of adjacent vertices in H is exactly 1 if H is a linear hypergraph. We get the following result.
For linear hypergraphs, the results in Corollary 2.3 are our main tools and we will use this technique in the sequel.

Main Results
The purpose of this section is to illustrate the use of the tools developed in Section 2, which translate nonspectral extremal problems into spectral results. We also determine bounds on the maximum spectral radius of uniform hypergraphs with girth at least five.

Spectral radius of linear hypergraphs without Fan k
To illustrate this technique, we first give a spectral version result corresponding to the linear Turán number of Fan k . We include this as a quick way for the reader to become acquainted with our notation.
For k ≥ 2, the k-fan Fan k is the k-uniform linear hypergraph with k edges f 1 , · · · , f k which pairwise intersect in the a single vertex v, and an additional edge g which intersects all f i in a vertex different from v. Füredi and Gyárfás studied the linear Turán number of Fan k in [10]. They proved that ex lin k (n, Fan k ) ≤ n 2 k 2 , asymptotically a factor of 1 + 1/(k − 1) smaller than the maximum number of edges in a k-uniform linear hypergraph, n 2 / k 2 . The Turán number of Fan k on k-uniform hypergraphs was determined by Mubayi and Pikhurko in [19]. A transversal design T (n, k) on n vertices with k groups is a k-partite hypergraph with groups of equal size (thus n is a multiple of k) and each pair of vertices from different groups is contained in exactly one hyperedge. Such designs have long been known to exist for all sufficiently large n (as a function of k), due to their connection with mutually orthogonal latin squares (MOLS). After adopting some notations and results of Füredi and Gyárfás, we prove the spectral analog of Füredi and Gyárfás's result and obtain the following extremal spectral result. Theorem 3.2 also implies that the maximum spectral radius of the adjacency tensor of hypergraphs is closely related to the Turán numbers of hypergraphs.
Theorem 3.2. Let H denote the set of linear k-uniform hypergraphs of order n, n ≡ 0 (mod k), with forbidden Fan k and ρ be the maximum spectral radius of hypergraphs in H. For n sufficiently large, we have ρ = n k .
Proof. Let H = (V, E) be a k-uniform Fan k -free linear hypergraph on n vertices. Set B u = V \ N u , where N u is the neighborhood of the vertex u. Suppose f ∈ E k (N u ); then, because H is linear, the vertices {v i } k i=1 of e must belong to k distinct edges {e i } k i=1 containing u. But then, {f, e 1 , . . . , e k } is the edge set of a Fan k , a contradiction. In other words, e k (N u ) = 0. On one hand, by Corollary 2.3, we have By Theorem 3.1, we know that |E(H)| ≤ n 2 k 2 . Thus, e(N u , B u ) ≤ n 2 k 2 , and using (12), we have ρ 2 ≤ n 2 k 2 , i.e., ρ ≤ n k . On the other hand, since a transversal design T (n, k) is an n k -regular linear hypergraph without Fan k , then ρ(T (n, k)) = n k . Since ρ is the maximum spectral radius of hypergraphs in H, we have ρ ≥ n k . This completes the proof.

Spectral radius of linear hypergraphs without Berge C 4
One of the first results concerning Turán numbers of Berge cycles is due to Lazebnik and Verstraëte [17]. Very recently this was strengthened by Ergemlidze, Győri and Methuku [7] who showed that ex lin 3 (n, {C 4 }) ≤ 1 6 n 3 2 + O(n) (which is tight, due to a construction from [17]). In this subsection, we give a spectral version of Ergemlidze, Győri and Methuku's results on k-uniform linear hypergraphs.
Our plan is to first upper bound u∈V v∈Nu d(v). After that, using the fundamental inequality ρ ≥ km/n (see [5]), we immediately obtain an upper bound on the number of edges as well. As before, let H be a linear hypergraph of order n and size m containing no Berge C 4 . To prove Theorem 3.3, we only need to upper bound v∈Nu d(v) by Corollary 2.3. Notice that d(v) = k t=1 e v t (N u ) for any v ∈ N u . Our plan is to estimate via an upper bound on e v t (N u ) for 1 ≤ t ≤ k.
with ω ∈ l} for any x ∈ N u , i.e., the set of vertices contained in some edge which intersects N u only at x. Note that S x ⊂ N x \ (N u ∪ {u}). The following results are necessary to our proof. Proof. Since x and y are adjacent, let l xy be the edge containing both the vertices x and y. Suppose for the sake of a contradiction that |N xy | ≥ 2k − 2.
Then there are at least k vertices in N xy other than the common neighbors in l xy . Since H is k-uniform, there must exist two distinct vertices v 1 and v 2 in N xy such that the pairs xv 1 and xv 2 are contained in two distinct edges which are not the edge l xy .
If the pairs yv 1 and yv 2 are contained in one edge incident to y, there must exist a vertex v 3 in N xy such that yv 1 and yv 3 are contained in two distinct edges incident to y. Then either the edges containing xv 1 , yv 1 , yv 3 and xv 3 or the edges containing xv 2 , yv 2 , yv 3 and xv 3 form a Berge C 4 in H, a contradiction. Otherwise the pairs yv 1 and yv 2 are contained in two different edges incident to y, then the four edges containing the pairs xv 1 , xv 2 , yv 1 and yv 2 form a Berge C 4 in H, a contradiction. Proof. Let l x , l y be two edges incident to u such that x ∈ l x and y ∈ l y . Since H is linear, it is obvious that l x = l y . If there were a vertex v in S x ∩ S y , then each of the pairs xv and yv would be contained in edges l x and l y so that l x ∩ N u = {x} and l y ∩ N u = {y}, whence l x = l y and neither edge contains u. But then the four edges l x , l y , l x , and l y are a Berge C 4 , a contradiction.
Proof of Theorem 3.3. First we show that k t=2 e v t (N u ) is no more than k for any v ∈ N u . Suppose for a contradiction that there is a vertex v ∈ N u such that k t=2 e v t (N u ) ≥ k + 1. Since H is linear, there is exactly one of these at least k + 1 edges containing u. Let l i , 1 ≤ i ≤ k, denote k other edges containing v. Since |l i ∩ N u | ≥ 2, we select k distinct vertices Then for the adjacent vertices u and v we have Since H is linear and For each 1 ≤ i ≤ d(u), let l i denote the i-th edge incident to u, and write Note that By Lemma 3.4, we have since the vertices p and q are adjacent. Then Thus we have and Since Combining this with (13), then The first inequality follows We now estimate v∈Nu (k − 1)d(v) in two different ways. On one hand, whered is the average degree of H. On the other hand, applying Cauchy-Schwarz, we immediately get Finally, combining (19) and (20), we have (k−1) 2d2 ≤ n+dk(k−1)(3k−5)/2. Solving this inequality, we getd ≤ ( k 2 (3k − 5) 2 + 16n + k(3k − 5))/(4(k − 1)). After applying this to (19), we obtain u∈V v∈Nu By Corollary 2.3, summing over all vertices in H, we get That is, ρ ≤ √ n k−1 + O(1). This completes the proof. Like Turán's theorem in extremal graph theory, Theorem 3.3 can serve as an entrée into related results. To begin with, note that Theorem 3.3 is closely related to hypergraph Turán numbers: indeed, the well-known inequality ρ ≥ km/n, immediately implies m ≤ n 3/2 k(k−1) + O(n). In other words, we showed that ex lin k (n, C 4 ) ≤ n 3/2 k(k−1) + O(n). Note that setting k = 3, we recover the results of Ergemlidze, Győri and Methuku [7]. Furthermore, since this result is known to be tight, it follows that Theorem 3.3 is tight as well. However, for k > 3, these results are not known to be tight, even in the exponent of n.
Moreover, our results answer an instance of the following broader question: Which subhypergraphs are necessarily present in a hypergraph H of sufficiently large order n if ρ − √ n k−1 → ∞?

Spectral radius of linear hypergraphs with at least girth five
The question we consider in this subsection is to determine the maximum spectral radius of k-uniform linear hypergraphs on n vertices of girth at least five. While the result below is a corollary of Theorem 3.3, we include its proof for two reasons: it is substantially simpler and shorter than the above proof, and the result holds independent interest because the maximum number of edges in graphs of girth five is an old problem of Erdős ([6]). Lazebnik and Verstraëte [17] showed that ex lin 3 (n, {C 3 , C 4 }) = n 3/2 /6 + O(n). Very recently this was strengthened by Ergemlidze, Győri, and Methuku, who showed in [7] that ex lin 3 (n, {C 3 , C 4 }) ∼ ex lin 3 (n, {C 4 }).
Theorem 3.6. Let H denote the set of linear k-uniform hypergraphs of order n with girth at least five and ρ be the maximum spectral radius of hypergraphs in H. For n sufficiently large, we have ρ ≤ √ n k−1 + O(1). Proof. Let H be a linear hypergraphs of order n and size m with girth at least five. It is easy to verify that E v k−1 (N u ) is exactly the set of edges incident to u and where v is an arbitrary vertex in N u . Otherwise H contains a Berge C 3 , a contradiction. That is to say k t=2 te t (N u ) = (k − 1)d(u). Now we only need to provide an upper bound on e 1 (N u ).
As before, set B u = V \{N u ∪{u}} and S v = {ω ∈ B u | ∃l ∈ E v 1 (N u ) with ω ∈ l} for any v ∈ N u . By Lemma 3.5, it is easy to verify that S x ∩ S y = ∅ for an arbitrary pair x, y in N u if x and y are nonadjacent. If x and y are adjacent, suppose for a contradiction that there is a vertex z ∈ S x ∩ S y . Let l xy be the edge containing both the vertices x and y. Similarly for l xz and l yz . Since H is linear, it is clear that l xy , l xz and l yz form a Berge C 3 , a contradiction. Thus e 1 (N u ) ≤ (n − (k − 1)d(u))/(k − 1). By Corollary 2.3, we get ρ 2 ≤ 1 k − 1 [e 1 (N u ) + 2e 2 (N u ) + · · · + ke k (N u )] ≤ n (k − 1) 2 + (k − 2)d(u) k − 1 .
Applying this to (25), we obtain that That is ρ ≤ √ n k−1 + O(1). This completes the proof.

Conclusion and open problems
Of course, the above results are just a small sample of spectral analogues for Turán-type hypergraph problems. Spectral extremal problems can be a rich font of interesting questions, since so many classical extremal problems have been investigated. See [15] for much more along these lines. Here we outline a few related to the above questions that we find particularly appealing.
and φ(f ) intersect exactly in φ(e ∩ f ) for every e, f ∈ E. In other words, F n consists of those linear 3-uniform hypergraphs whose edges can be faithfully embedded as planar triangles in R 3 . Then, if f (n) is the maximum number of edges of any hypergraph in F n , is it true that f (n) = o(n 2 )? Károlyi and Solymosi showed in [14] that f (n) = Ω(n 3/2 ), but not much more is known about this question. We ask the spectral analogue: for H ∈ F n , is it true that ρ(H) = o(n)?