Extremal problems for the p-spectral radius of graphs

The $p$-spectral radius of a graph $G\ $of order $n$ is defined for any real number $p\geq1$ as \[ \lambda^{\left( p\right) }\left( G\right) =\max\left\{ 2\sum_{\{i,j\}\in E\left( G\right) \ }x_{i}x_{j}:x_{1},\ldots,x_{n}\in\mathbb{R}\text{ and }\left\vert x_{1}\right\vert ^{p}+\cdots+\left\vert x_{n}\right\vert ^{p}=1\right\} . \] The most remarkable feature of $\lambda^{\left( p\right) }$ is that it seamlessly joins several other graph parameters, e.g., $\lambda^{\left( 1\right) }$ is the Lagrangian, $\lambda^{\left( 2\right) }$ is the spectral radius and $\lambda^{\left( \infty\right) }/2$ is the number of edges. This paper presents solutions to some extremal problems about $\lambda^{\left( p\right) }$, which are common generalizations of corresponding edge and spectral extremal problems. Let $T_{r}\left( n\right) $ be the $r$-partite Tur\'{a}n graph of order $n.$ Two of the main results in the paper are: (I) Let $r\geq2$ and $p>1.$ If $G$ is a $K_{r+1}$-free graph of order $n,$ then \[ \lambda^{\left( p\right) }\left( G\right)<\lambda^{\left( p\right) }\left( T_{r}\left( n\right) \right) , \] unless $G=T_{r}\left( n\right) .$ (II) Let $r\geq2$ and $p>1.$ If $G\ $is a graph of order $n,$ with \[ \lambda^{\left( p\right) }\left( G\right)>\lambda^{\left( p\right) }\left( T_{r}\left( n\right) \right) , \] then $G$ has an edge contained in at least $cn^{r-1}$ cliques of order $r+1,$ where $c$ is a positive number depending only on $p$ and $r.$


Introduction
In this paper we study extremal problems for the p-spectral radius λ (p) of graphs, so first let us recall the definition of λ (p) . Suppose that G is a graph of order n. The quadratic form of G is defined for any vector [x i ] ∈ R n as Suppose that H is a graph which is necessary present in any graph G of sufficiently large order n, with λ (p) (G) > λ (p) (T r (n)). What is the structure of a graph G of order n if λ (p) (G) > λ (p) (T r (n)) − o n 2−2/p , but G contains no H?
Many extremal problems along the above lines have been successfully solved for λ (2) , the classical spectral radius; see [17] for a survey and references. However, λ (2) belongs to the realm of Linear Algebra and its study builds on proven solid ground. By contrast, linear-algebraic methods are irrelevant for the study of λ (p) in general, and in fact no efficient general methods are known for it. Thus the study of λ (p) for p = 2 is far more complicated than of λ (2) . One of the aims on the present study is to find out if specific applications of the spectral radius λ (2) can be extended to λ (p) in general. So far, most attempts have been successful, but there are many basic unanswered questions, see [20] for some examples.
It should be noted that extremal problems for λ (p) of hypergraphs have been studied in [9], [10], [19], and [20], but 2-graphs are better understood, so it is worthwhile to delve into deeper extremal theory. Another line has been investigated in [18], where the emphasis is on hereditary properties.

Turán type theorems for λ (p) (G)
It is not hard to see that if n ≥ r > q, then λ (p) (T r (n)) > λ (p) (T q (n)) for every p ≥ 1. This observation entails the following reformulation of Theorem 1.
Let us note that particular relations between the clique number ω of a graph G and λ (p) (G) have been long known. For example, the result of Motzkin and Straus [11] (see Theorem E below) establishes the fundamental fact that λ (1) (G) = 1 − 1/ω; later it has been used by Wilf [23] and in [12] it was used for the stronger inequality Note that the last two results are explicit, while being almost tight. It turns out that the approach of Motzkin and Straus helps to deduce similar explicit results for λ (p) (G) and any p ≥ 1 as well.
Theorem 3 Let r ≥ 2 and p ≥ 1. If G is a K r+1 -free graph of order n, then and If p > 1, equality holds in (2) if and only if r|n and G = T r (n) .
In particular, Theorem 3 implies that if G is a graph of order n, with clique number ω, then A natural question is how good bounds (1) and (2) are compared to the bound in Theorem 1, which is attained for every n. It turns out that bounds (1) and (2) are never too far from the best possible one, as seen in the following several estimates.
Theorem 4 Let T r (n) be the r-partite Turán graph of order n. Then and for every p > 1, 3 Saturation problems Theorem 1 implies that if G is a graph of order n, with λ (p) (G) > λ (p) (T r (n)) , then G contains a K r+1 . We shall show that, in fact, much larger supergraphs of K r+1 can be found in G. Such problems are usually called saturation problems.

Joints
In [5] Erdős proved that if r ≥ 2, and G is a graph of sufficiently large order n, with e (G) > e (T r (n)) , then G has an edge that is contained in at least n r−1 / (10r) 6r cliques of order r + 1. This fact is fundamental, so to study its consequences, the following definition was given in [3]: An r-joint of size t is a collection of t distinct r-cliques sharing an edge.
A 3-joint is also called a book. Books have been studied extensively in extremal and Ramsey graph theory. Note that books are determined by their size alone, while for r > 3 there are many non-isomorphic r-joints of the same size.
We write js r (G) for the maximum size of an r-joint in a graph G. The following theorem enhances Theorem 1, insofar that from the same premises it implies the existence of subgraphs whose order grows with n. For this reason we shall use it as a starting point for several other extensions.
Theorem 5 Let r ≥ 2 and p > 1. If G is a graph of order n, with Let us note that the order of n r−1 is obviously best possible, but the coefficient r −r 6 p/(p−1) is far from being optimal. Nevertheless, this small coefficient makes the statement valid for all n, and for larger n it can be somewhat increased.

Color critical subgraphs
Call a graph k-color critical, if it is k-colorable, but it can be made (k − 1)-colorable by removing a particular edge. For example, books are 3-color critical graphs.
Simonovits [21] has proved that if F is an (r + 1)-color critical graph, then F ⊂ G for every graph G of sufficiently large order n, with e (G) > e (T r (n)).
This statement can be generalized considerably. Indeed, given the integers r ≥ 2 and t ≥ 2, let K + r (t) be the complete r-partite graph with each part of size t, and with an edge added to its first part. The study of K + r (t) in connection to the Turán theorem has been initiated by Erdős [4], [6], but a definite result has been obtained only in [16]: Theorem C Let r ≥ 2 and c ≤ c 0 (r) be a sufficiently small positive number. If G is a graph of sufficiently large order n, with e (G) > e (T r (n)), then G contains a K + r (⌊c log n⌋) . This type of result is indeed a neat generalization of Simonovits's result, for any (r + 1)-color critical graph is a subgraph of K + r (⌊c log n⌋) if n is large enough. In [15] a similar theorem has been proved also for the spectral radius λ (2) : Theorem D Let r ≥ 2 and c ≤ c 0 (r) be sufficiently small positive number. If G is a graph of sufficiently large order n, with λ (2) (G) > λ (2) (T r (n)), then G contains a K + r (⌊c log n⌋) . We give a common generalization of Theorems C and D in the following theorem.
Theorem 6 Let r, p, c, and n satisfy Let us emphasize that in Theorem 6 c may depend on n, e.g., if c is a slowly decaying function of n, like c = 1/ log log n, the conclusion is meaningful for sufficiently large n.
It should be noted that the authors of [10], in their Corollary 2, prove a similar theorem, where instead of K + r (⌊c log n⌋) they take a fixed (r + 1)-color critical subgraph. However, they claim that their statement generalizes Theorem D as well, which is false, as the order of K + r (⌊c log n⌋) grows with n. In fact the change from a fixed (r + 1)-color critical graph to K + r (⌊c log n⌋) is a major difference, requiring a longer proof, with more advanced techniques and more delicate calculations.

An abstract saturation theorem
The proofs of Theorems 5 and 6, and of several stability results in a forthcoming paper, will be deduced from a fairly general, multiparameter statement stated as follows.
Theorem 7 Let the numbers p, γ, A, R, and n satisfy If G is a graph of order n, with This theorem seems overly complicated, but its meaning and usage are straightforward. It will be applied to prove the existence of certain subgraphs. The starting point will be some known statement ensuring that if G is a graph of sufficiently large order n, with , and δ (G) ≤ (A − γ) n, so the requirement for the existence of F are not met at all. In this case Theorem 7 helps to mend the situation, as it guarantees that there is an induced subgraph H ⊂ G of relatively large order k, satisfying Now, if n is large enough, then k is large enough, and so F ⊂ H ⊂ G, as desired.
Let us note that, in any concrete case, the choice of γ, A and R is determined by the type of the subgraph F.
In the remaining part of the paper we prove Theorems 1-7.

Notation and preliminaries
In our proofs we shall use a number of classical inequalities: the Power Mean inequality (PM inequality), the Bernoulli and the Maclaurin inequalities; for more details on these inequalities we refer the reader to [7].
For graph notation and concepts undefined here, we refer the reader to [2]. In particular, given a graph G, we write: -V (G) for the vertex set of G and v (G) for |V (G)| ; -E (G) for the edge set of G and e (G) for |E (G)| ; -Γ G (u) for the set of neighbors of a vertex u (we drop the subscript if G is understood); -δ (G) for the minimum degree of G; -k r (G) for the number of r-cliques of G; -G − u for the graph obtained by removing the vertex u ∈ V (G) .
If G is a graph of order n and V (G) is not defined explicitly, it is assumed that V (G) := {1, . . . , n} .

Some facts about the p-spectral radius
All required facts about the p-spectral radius of graphs are given below. Additional reference material can be found in [9], [19], and [20].
Let G be a graph of order n. A vector [x i ] ∈ R n such that |x 1 | p + · · · + |x n | p = 1 and It is easy to see,that there is always a non-negative eigenvector to λ (p) (G) . If p > 1, by Lagrange's method, one can show that for each k = 1, . . . , n. Equation (6) is called the eigenequation of λ (p) (G) for the vertex k.
In the following three bounds it is assumed that p ≥ 1. First, by Maclaurin's and the PM inequalities we find the absolute maximum of λ (p) (G) with respect to n : Second, we find a bound with respect to e (G) : In the other direction, taking the n-vector x = n −1/p , . . . , n −1/p , we obtain a useful lower bound Note that if 1 ≤ p < 2, then bound (9) may not be tight for some regular graphs, but for p ≥ 2 it is always tight for regular graphs; in fact, as mentioned earlier, It is worth noting that using the PM inequality, one can find that λ (p) (G) n 2/p is nonincreasing in p, that is to say, if p > q ≥ 1, then

Proof of Theorem 1
Since a statement similar to Theorem 1 has been claimed in [10], Corollary 2, we need to make a comment here. The proof given below reduces Theorem 1 to r-partite graphs, for which we already gave an independent proof in [9]. The same reduction, albeit more complicated, has been carried out in [10] as well, but these authors provide no proof for r-partite graphs, so their proof of Theorem 1 is essentially incomplete. Unfortunately, this omission is not negligible, as the proof for r-partite graphs is much longer and more involved than the reduction of Theorem 1 to r-partite graphs. Next, we state the main ingredient of our proof, which is a particular instance of a result in [9] about the p-spectral radius of k-partite uniform hypergraphs.
Theorem 8 Let r ≥ 2, and p > 1. If G is an r-partite graph of order n, then Thus, to prove Theorem 1, all we need is that the maximum λ (p) (G) of a K r+1 -free graph G of order n is attained on an r-partite graph. Reductions of this kind have been pioneered by Zykov [24] and Erdős, but to spectral problems they have been first applied by Guiduli, in an unpublished proof of the spectral Turán theorem. Another noteworthy application of the same techniques is for the spectral radius of the signless Laplacian of K r+1 -free graphs in [8]. Thus we proceed with a reduction lemma for λ (p) (G) of a K r+1 -free graph G.
Lemma 9 Let p ≥ 1. If G is a K r+1 -free graph of order n, then there exists an r-partite graph H of order n such that λ (p) (H) ≥ λ (p) (G) .
Proof Let x = [x i ] be a nonnegative eigenvector to λ (p) (G) . For each v ∈ V (G) , set We shall prove that there exists a complete r-partite graph H such that V (H) = V (G) and D H (v, x) ≥ D G (v, x) for any v ∈ V (G) . This proof will be carried out by induction on r. Let and set U := Γ G (u) and W := V (G) \Γ G (u) . To start the induction let r := 2; hence G is triangle-free, and so e (G [U ]) = 0. We shall show that the complete bipartite graph H with bipartition V (H) = U ∪ W is as required. Indeed, if v ∈ U, then Γ G (v) ⊂ W, and so Hence the graph H is as required. Now, let r > 2 and assume that the assertion is true for r ′ whenever 2 ≤ r ′ < r. First note that G [U ] is a K r -free graph; hence, by the induction assumption there exists a complete (r − 1)-partite graph F with V (F ) = U and D F (v, x) ≥ D G[U ] (v, x) for any vertex v ∈ U. Let V (F ) = V 1 ∪ · · · ∪ V r−1 be the partition of V (F ) into independent sets and let H be the complete r-partite graph with partition We shall prove that H is as required. Indeed, on the one hand, if v ∈ U, then . This completes the induction step, and the existence of H is proved.
To finish the proof of the lemma, note that

Proofs of Theorems 3 and 4
We use below the result of Motzkin and Straus [11], that can be stated as: Theorem E If G is a K r+1 -free graph of order n, and x 1 , . . . , x n are nonnegative numbers such that x 1 + · · · + x n = 1, then The conditions for equality in (11) are well known, but we shall omit them. Instead we just note that if K r ⊂ G, one may choose a vector (x 1 , . . . , x n ) so that equality holds in (11).
We often shall use the following bound on the number of edges of the Turán graph T r (n) , Indeed, let n = rs + t, where s and t are nonnegative integers and 0 ≤ t < s. It is known that e (T r (n)) = r 2 hence, 2e (T r (n)) = 2 r 2 n 2 − t 2 r 2 + 2

Proof of Theorem 3
The proof of inequality (1) has been given many times, but it is short, so for reader's sake we shall give it again. Let [x i ] be a nonnegative eigenvector to λ (p) (G) . The PM inequality implies that Note that x p 1 + · · · + x p n = 1, and G is K r+1 -free, so the Motzkin-Straus result implies that Plugging this back in (13), we obtain (1).
To prove (2), we use (1) and the concise Turán theorem, which implies that Now, if equality holds, i.e., if and this can happen only if r|n and G = T r (n) .
Proof of Theorem 4 Equality (3) follows from the Motzkin-Straus' Theorem E and the fact the K r ⊂ T r (n) . The lower bound in (4) follows by (9). To prove the upper bound in (4) let us start with Next, using (12) and Bernoulli's inequality, we get the estimate This implies (2e (T r (n))) 1/p n 2/p 1 + r pn 2 > 1 −

Proof of Theorem 7
The proof of Theorem 7 goes along lines, which are familiar from Theorem 5 in [14], but the arguments and calculations are more complicated. To clarify the structure of the proof we have extracted two of its essential points into Lemmas 10 and 11.
Now, plugging the bounds on δ (G) and λ (p) (G) n 2/p−1 , we get To complete the proof of the lemma we shall check that Indeed, Lemma 10 is proved.
The next lemma shows that if λ (p) (G) n 2/p−1 is large enough, but the minimum degree δ (G) is not too large, we can remove a vertex u, so that λ (p) (G − u) (n − 1) 2/p−1 is also large.
Let G be a graph of order n, with If [x i ] is a nonnegative eigenvector to λ (p) (G) and u is a vertex with x u = min {x 1 , . . . , x n } , then Proof Let p, γ, A, R, and n satisfy the requirements, let x = [x i ] be a nonnegative eigenvector to λ (p) (G) and u be a vertex with x u := min {x 1 , . . . , x n } ; set σ := x p k . Obviously, Lemma 10 can be applied here, getting Next, set for short δ := δ (G) , λ n := λ (p) (G) , and λ n−1 := λ (p) (G − u) . Letting x ′ be the (n − 1)-vector obtained from x by omitting the entry x k , we see that On the other hand, hence, after some algebra, we find that Note that the function is decreasing in y for 0 < y < 1, for the derivative of f (y) satisfies Therefore, in view of (15), we find that Thus, setting for short ξ := γ/2, we see that Plugging this back in (16), we find that To estimate the latter expression, note that 0 < 1 − ξ < 1 and 0 < ξ < 1; hence, Bernoulli's inequality implies that Thus, we obtain as claimed. Lemma 11 is proved.
The main idea of the proof of Theorem 7 is to iterate the removal of vertices of smallest entry in eigenvectors to λ (p) . Every time a vertex is removed, the ratio of λ (p) of the remaining graph to its order increases. So the vertex removal must stop before λ (p) exceeds its absolute maximum. As this stop happens fairly soon, the order of the remaining graph is fairly large.
Proof of Theorem 7 Let p, γ, A, R and n be as required, and let G be a graph of order n, with λ (p) (G) n 2/p−1 > An − R/n and δ (G) ≤ (A − γ) n. and so, implying (17). Therefore, after removing the vertex u, Lemma 11, together with (5), implies that (18) holds as well. This completes the induction step and the proof of (i) and (ii). Finally, let H := G i and k := v (H) = i, where G i is the last graph generated by P. We shall prove the following three properties of H : k > A p/(γp−γ) n (21) Indeed, inequality (20) is obvious, as this is the loop exit condition. Also inequality (21) holds because of (17). Finally, note that To prove (22), we shall show that which is equivalent to Assume the latter inequality fails, that is to say, Using Bernoulli's inequality, we get After some rearrangement we obtain Now, obviously and also Therefore, (25) is a contradiction and (23) holds.
Hence the graph H has the required properties and Theorem 7 is proved.

Proof of Theorem 5
The proof of Theorem 5 is based on the following nonspectral result, proved in [3].
The idea of Lemma 12 can be traced back to Erdős; its main advantage is that the bound on the jointsize can be deduced from two simpler conditions: presence of K r+1 and sufficient minimum degree. Although, these conditions may not hold in G, Theorem 7 guarantees that there is a large subgraph H of G for which the conditions do hold. Now, applying Lemma 12 to H, we obtain the desired bound on js r+1 (G).

Proof of Theorem 6
For the proof of Theorem 6 we rely on Theorem 7 and on a non-spectral result, proved in [16], Theorem 6. To state it we first extend the definition of K + r (t) as follows: given the integers r ≥ 2, s ≥ 2, and t ≥ 1, let K + r (s; t) be the complete r-partite graph with first r − 1 parts of size s and the last part of size t, and with an edge added to its first part.
If G is a graph of order n, with K r+1 ⊂ G and δ (G) > 1 − 1/r − 1/r 4 n, then G contains a K + r ⌊c log n⌋ ; n 1−cr 3 . .
For the proof of Theorem 6 we shall need a corollary of this statement.
If G is a graph of order n, with K r+1 ⊂ G and δ (G) > 1 − 1/r − 1/r 4 n, then G contains a K + r (⌊a log n⌋) .
Proof There is not much to prove here. Indeed assume that r, a, and n are as required and let G be a graph of order n, with K r+1 ⊂ G and δ (G) > 1 − 1/r − 1/r 4 n. By Theorem 13, G contains a K + r ⌊a log n⌋ ; n 1−ar 3 . First, note that log n ≥ 2 a ≥ 2r (r+8)r > 2; hence n 1/2 > e and so, n 1/2 > log n 1/2 . Now, with a lot to spare, we see that n 1−ar 3 > n 1/2 > 1 2 log n > r −(r+8)r log n ≥ a log n.
So K + r (⌊a log n⌋) ⊂ K + r ⌊a log n⌋ ; n 1−ar 3 ⊂ G, completing the proof of Corollary 14.