Spectral Tur´an Type Problems on Cancellative Hypergraphs

Let G be a cancellative 3-uniform hypergraph in which the symmetric diﬀerence of any two edges is not contained in a third one. Equivalently, a 3-uniform hypergraph G is cancellative if and only if G is { F 4 , F 5 } -free, where F 4 = { abc, abd, bcd } and F 5 = { abc, abd, cde } . A classical result in extremal combinatorics stated that the maximum size of a cancellative hypergraph is achieved by the balanced complete tripartite 3-uniform hypergraph, which was ﬁrstly proved by Bollob´as and later by Keevash and Mubayi. In this paper, we consider spectral extremal problems for cancellative hypergraphs. More precisely, we determine the maximum p -spectral radius of cancellative 3-uniform hypergraphs, and characterize the extremal hypergraph. As a by-product, we give an alternative proof of Bollob´as’ result from spectral viewpoint.


Introduction
Consider an r-uniform hypergraph (or r-graph for brevity) G and a family of r-graphs F. We say G is F-free if G does not contain any member of F as a subhypergraph.The Turán number ex(n, F) is the maximum number of edges of an F-free hypergraph on n vertices.Determining Turán numbers of graphs and hypergraphs is one of the central problems in extremal combinatorics.For graphs, the problem was asymptotically solved for all nonbipartite graphs by the celebrated Erdős-Stone-Simonovits Theorem.By contrast with the graph case, there is comparatively little understanding of the hypergraph Turán number.We refer the reader to the surveys [6,9,12].
In this paper we consider spectral analogues of Turán type problems for r-graphs.For r = 2, the picture is relatively complete, due in large part to a longstanding project of Nikiforov, see e.g., [13] for details.However, for r ≥ 3 there are very few known results.In [10], Keevash-Lenz-Mubayi determine the maximum p-spectral radius of any 3-graph on n vertices not containing the Fano plane when n is sufficiently large.They also obtain a p-spectral version of the Erdős-Ko-Rado theorem on t-intersecting r-graphs.Recently, Ellingham-Lu-Wang [4] show that the n-vertex outerplanar 3-graph of maximum spectral radius is the unique 3-graph whose shadow graph is the join of an isolated vertex and the path P n−1 .Gao-Chang-Hou [7] study the extremal problem for K + r+1 -free r-graphs among linear hypergraphs, where K + r+1 is obtained from the complete graph K r+1 by enlarging each edge of K r+1 with r−2 new vertices disjoint from V (K r+1 ) such that distinct edges of K r+1 are enlarged by distinct vertices.
To state our results precisely, we need some basic definitions and notations.A 3-graph is tripartite or 3-partite if it has a vertex partition into three parts such that every edge has exactly one vertex in each part.Let T 3 (n) be the complete 3-partite 3-graph on n vertices with part sizes ⌊n/3⌋, ⌊(n + 1)/3⌋, ⌊(n + 2)/3⌋, and t 3 (n) be the number of edges of T 3 (n).That is, We call an r-graph G cancellative if G has the property that for any edges A, B, C whenever where △ is the symmetric difference.For graphs, the condition is equivalent to saying that G is triangle-free.Moving on to 3-graphs, we observe that B∆C ⊂ A can only occur when |B ∩ C| = 2 for B = C.This leads us to identify the two non-isomorphic configurations that are forbidden in a cancellative 3-graph: F 4 = {abc, abd, bcd} and F 5 = {abc, abd, cde}.
It is well-known that the study of Turán numbers dates back to Mantel's theorem, which states that ex(n, K 3 ) = ⌊n 2 /4⌋.As an extension of the problem to hypergraphs, Katona conjectured, and Bollobás [1] proved the following result.
In [8], Keevash and Mubayi presented a new proof of Bollobás' result, and further proved a stability theorem for cancellative hypergraphs.The main result of this paper is the following p-spectral analogues of Bollobás' result.
Theorem 1.2.Let p ≥ 1 and G be a cancellative 3-graph on n vertices.

Preliminaries
In this section we introduce definitions and notation that will be used throughout the paper, and give some preliminary lemmas.
Given an r-graph G = (V (G), E(G)) and a vertex v of G.The link L G (v) is the (r − 1)graph consisting of all S ⊂ V (G) with |S| = r − 1 and S ∪ {v} ∈ E(G).The degree d G (v) of v is the size of L G (v).As usual, we denote by N G (v) the neighbor of a vertex v, i.e., the set formed by all the vertices which form an edge with v.In the above mentioned notation, we will skip the index G whenever G is understood from the context.
The shadow graph of G, denoted by ∂(G), is the graph with V (∂(G)) = V (G) and E(∂(G)) consisting of all pairs of vertices that belong to an edge of G, i.e., E(∂(G)) = {e : |e| = 2, e ⊆ f for some f ∈ E(G)}.For more definitions and notation from hypergraph theory, see e.g., [2].
For any real number p ≥ 1, the p-spectral radius was introduced by Keevash, Lenz and Mubayi [10] and subsequently studied by Nikiforov [14,15].Let G be an r-graph of order n, the polynomial form of G is a multi-linear function P G (x) : R n → R defined for any vector The p-spectral radius1 of G is defined as where For any real number p ≥ 1, we denote by S n−1 p,+ the set of all nonnegative real vectors x ∈ R n with x p = 1.If x ∈ R n is a vector with x p = 1 such that λ (p) (G) = P G (x), then x is called an eigenvector corresponding to λ (p) (G).Note that P G (x) can always reach its maximum at some nonnegative vectors.By Lagrange's method, we have the eigenequations for λ (p) (G) and x ∈ S n−1 p,+ as follows: It is worth mentioning that the p-spectral radius λ (p) (G) shows remarkable connections with some hypergraph invariants.For instance, λ (1) (G)/r is the Lagrangian of G, λ (r) (G) is the usual spectral radius introduced by Cooper and Dutle [3], and λ (∞) (G)/r is the number of edges of G (see [14,Proposition 2.10]).
Given two vertices u and v, we say that u and v are equivalent in G, in writing u ∼ v, if transposing u and v and leaving the remaining vertices intact, we get an automorphism of G.

Lemma 2.1 ([14]
).Let G be a uniform hypergraph on n vertices and u ∼ v.If p > 1 and

Cancellative hypergraph of maximum p-spectral radius
The aim of this section is to give a proof of Theorem 1.2.We split it into Theorem 3.1 -Theorem 3.3, which deal with p = 3, p > 3 and p = 1, respectively.

General properties on cancellative hypergraphs
We start this subsection with a basic fact.Lemma 3.1.Let G be a cancellative hypergraph, and u, v be adjacent vertices.Then L(u) and L(v) are edge-disjoint graphs.
Proof.Assume by contradiction that e ∈ E(L(u)) ∩ E(L(v)).Since u and v are adjacent in G, we have {u, v} ⊂ e 1 ∈ E(G) for some edge e 1 .Hence, e 2 = e ∪ {u}, e 3 = e ∪ {v} and e 1 are three edges of G such that e 2 ∆e 3 ⊂ e 1 , a contradiction.
Let G be a 3-graph and v ∈ V (G).We denote by and Proof.Suppose to the contrary that there exist three edges Let x ∈ S n−1 p,+ be an eigenvector corresponding to λ (p) (G).By Lemma 2.1, for i = 1, 2, 3 we denote a i := x v for v ∈ V i , and set λ := λ (p) (G) for short.In light of eigenequation (2.2), we find that from which we obtain that a i = (3n i ) −1/p , i = 1, 2, 3. Therefore, This completes the proof of Lemma 3.3.

Extremal p-spectral radius of cancellative hypergraphs
Let Ex sp (n, {F 4 , F 5 }) be the set of all 3-graphs attaining the maximum p-spectral radius among cancellative hypergraphs on n vertices.Given a vector x ∈ R n and a set S ⊂ [n] := {1, 2, . . ., n}, we write x(S) := i∈S x i for short.The support set S of a vector x is the index of non-zero elements in x, i.e., S = {i ∈ [n] : x i = 0}.Also, we denote by x(e) − 3 e∈Eu(G) x(e) + 3 x(e \ {v}) which yields that x u ≥ x v .Likewise, we also have and u, v be two non-adjacent vertices.Then there exists a cancellative 3-graph H such that Proof.Assume that x ∈ S n−1 p,+ is an eigenvector corresponding to λ (p) (G).By Lemma 3.4, x u = x v .Without loss of generality, we assume d G (u) ≥ d G (v).In view of (2.1) and (2.2), we have x(e) − 3 e∈Eu(G) x(e) + 3 e∈Ev(G) x(e \ {v}) Observe that T v u (G) is a cancellative 3-graph and G ∈ Ex sp (n, {F 4 , F 5 }).We immediately obtain that λ (p) (T v u (G)) = λ (p) (G).It is straightforward to check that H := T v u (G) is a cancellative 3-graph satisfying (3.1), as desired.
Next, we give an estimation on the entries of eigenvectors corresponding to λ (p) (G).Lemma 3.6.Let G ∈ Ex sp (n, {F 4 , F 5 }) and x ∈ S n−1 p,+ be an eigenvector corresponding to • x max .
Proof.Suppose to the contrary that x min ≤ 3 4 2/(p−1) • x max .Let u and v be two vertices such that x u = x min and x v = x max > 0. Then we have On the other hand, by eigenequations we have e∈Ev(G)\Eu(G) x(e) ≥ λ (p) (G)(x p v − x p u ).
(3.3) Now, we consider the cancellative 3-graph T v u (G).In light of (2.1) and (3.3), we have x(e) − 3 e∈Eu(G) x(e) + 3 x(e \ {v}) where the third inequality is due to (3.2).This contradicts the fact that G has maximum p-spectral radius over all cancellative hypergraphs.Now, we are ready to give a proof of Theorem 1.2 for p = 3. Theorem 3.1.Let G be a cancellative 3-graph on n vertices.Then λ (3) Proof.According to Lemma 3.5, we assume that G * ∈ Ex sp (n, {F 4 , F 5 }) is a 3-graph such that L G * (u) = L G * (v) for any non-adjacent vertices u and v.
Our first goal is to show Recall that for any non-adjacent vertices u and v we have Hence, the sets U 1 , U 2 and U 3 are well-defined.
Claim 3.1.The following statements hold: (1) Proof of Claim 3.1.Since T 3 (n) is a cancellative 3-graph, it follows from Lemma 3.3 that By simple algebra we see (3.4) (1).By eigenequation with respect to u 1 , we have Combining with (3.4), we get (2).Observe that the definition of U 1 , and which, together with Lemma 3.6 for p = 3, gives The last inequality is due to (3.4).
(3).Let v be an arbitrary vertex in V (G * ).Then Hence, by Lemma 3.6 and (3.4) we have as desired.
On the other hand, since . Therefore, every pair of vertices in {v 1 , v 2 , v 3 , v 4 , u 1 , u 2 , u 3 } is contained in an edge of G * .Consider the graph By Claim 3.1, we have a contradiction completing the proof of Claim 3.2.
Proof of Claim 3.3.Suppose to the contrary that L = ∅.For i = 1, 2, 3, let L i be the set of vertices in L which is not contained in an edge with coloring i.By Claim 3.2, we have Without loss of generality, we assume L 1 = ∅.Let w be a vertex in L 1 .
Then there exists an edge f in G * such that f = {u 1 , w, w ′ }, where This implies that w ∈ U 3 , a contradiction to w ∈ L. Similarly, if w ′ ∈ U 3 , then w ∈ U 2 , which is also a contradiction.Now, we continue our proof.By Claim 3.3, we immediately obtain that G * is a complete 3-partite 3-graph with vertex classes U 1 , U 2 and U 3 .Hence, G * = T 3 (n) by Lemma 3.3.
Finally, it is enough to show that G = T 3 (n) for any G ∈ Ex sp (n, {F 4 , F 5 }).According to Lemma 3.5 and Claim 3.3, we can transfer G to the complete 3-partite 3-graph T 3 (n) by a sequence of switchings T v u ( • ) that keeping the spectral radius unchanged.Let T 1 , . . ., T s be such a sequence of switchings be an eigenvector corresponding to λ (3) (G s−1 ) and T v u (G s−1 ) = T 3 (n), and denote , and therefore G = T 3 (n).This completes the proof of the theorem.
According to Theorem 3.1, we can give an alternative proof of Bollobás' result for n ≡ 0 (mod 3).Proof.Denote by z the all-ones vector of dimension n.In view of (2.1), we deduce that On the other hand, by Theorem 3.1 we have Equality may occur only if λ (3) , and therefore G = T 3 (n) by Theorem 3.1.
Next, we will prove Theorem 1.2 for the case p > 3 as stated in Theorem 3.2.On the other hand, we have We immediately obtain m ≥ t 3 (n).The result follows from Theorem 1.1.
Finally, we shall give a proof of Theorem 1.2 for the remaining case p = 1.In what follows, we always assume that x ∈ S n−1 1,+ is an eigenvector such that x has the minimum possible number of non-zero entries among all eigenvectors corresponding to λ (1) (G).Before continuing, we need the following result.Lemma 3.8 ([5]).Let G be an r-graph and S be the support set of x.Then for each pair vertices u and v in S, there is an edge in G[S] containing both u and v. Theorem 3.3.Let G be a cancellative 3-graph.Then λ (1) (G) = 1/9.

Lemma 3 . 4 .
Let p > 1, G ∈ Ex sp (n, {F 4 , F 5 }), and x ∈ S n−1 p,+ be an eigenvector corresponding to λ (p) (G).If u, v are two non-adjacent vertices, then x u = x v .Proof.Assume u and v are two non-adjacent vertices in G. Since G is a cancellative 3-graph, we have T v u (G) is also cancellative by Lemma 3.2.It follows from (2.1) and (2.2) that