key: cord-0146228-m9hrepou
authors: Li, Yongtao; Peng, Yuejian
title: Refinement on spectral Tur'{a}n's theorem
date: 2022-04-20
journal: nan
DOI: nan
sha: d7c5b5758f59b558783f183a15962cc0219fddda
doc_id: 146228
cord_uid: m9hrepou

A well-known result in extremal spectral graph theory, known as Nosal's theorem, states that if $G$ is a triangle-free graph on $n$ vertices, then $lambda (G) le lambda (K_{lfloor frac{n}{2}rfloor, lceil frac{n}{2} rceil })$, equality holds if and only if $G=K_{lfloor frac{n}{2}rfloor, lceil frac{n}{2} rceil }$. Nikiforov [Linear Algebra Appl. 427 (2007)] extended Nosal's theorem to $K_{r+1}$-free graphs for every integer $rge 2$. This is known as the spectral Tur'{a}n theorem. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] proved a refinement on Nosal's theorem for non-bipartite triangle-free graphs. In this paper, we provide alternative proofs for the result of Nikiforov and the result of Lin, Ning and Wu. Our proof can allow us to extend the later result to non-$r$-partite $K_{r+1}$-free graphs. Our result refines the theorem of Nikiforov and it also can be viewed as a spectral version of a theorem of Brouwer.

Extremal graph theory is becoming one of the significant branches of discrete mathematics nowadays, and it has experienced an impressive growth during the last few decades. With the rapid development of combinatorial number theory and combinatorial geometry, extremal graph theory has a large number of applications to these areas of mathematics. Problems in extremal graph theory deal usually with the question of determining or estimating the maximum or minimum possible size of graphs satisfying some certain requirements, and further characterize the extremal graphs attaining the bound. For example, one of the most well-studied problems is the Turán-type problem, which asks to determine the maximum number of edges in a graph which forbids the occurence of some specific substructures. Such problems are related to other areas including theoretical computer science, discrete geometry, information theory and number theory.

Given a graph F , we say that a graph G is F -free if it does not contain an isomorphic copy of F as a subgraph. For example, every bipartite graph is C 3 -free, where C 3 is a triangle. The Turán number of a graph F is the maximum number of edges in an F -free n-vertex graph, and it is usually denoted by ex(n, F ). An F -free graph on n vertices with ex(n, F ) edges is called an extremal graph for F . Over a century old, a well-known theorem of Mantel [35] states that every n-vertex graph with more than n 2 4 edges must contain a triangle as a subgraph. We denote by K s,t the complete bipartite graph with parts of sizes s and t.

Theorem 1.1 (Mantel, 1907) . Let G be an n-vertex graph. If G is triangle-free, then e(G) ≤ e(K n 2 , n 2 ) = n 2 4 , equality holds if and only if G = K n 2 , n 2 . Let K r+1 be the complete graph on r + 1 vertices. In 1941, Turán [48] proposed the natural question of determining ex(n, K r+1 ) for every integer r ≥ 2. Let T r (n) denote the complete r-partite graph on n vertices whose part sizes are as equal as possible. That is, T r (n) = K t 1 ,t 2 ,...,tr with r i=1 t i = n and |t i − t j | ≤ 1 for i = j. This implies that each vertex part of T r (n) has size either n r or n r . The graph T r (n) is usually called the Turán graph. In particular, we have T 2 (n) = K n 2 , n 2 . Importantly, Turán [48] extended Mantel's theorem and proved the following result. Theorem 1.2 (Turán, 1941) . Let G be a graph on n vertices. If G is K r+1 -free, then e(G) ≤ e(T r (n)), equality holds if and only if G is the r-partite Turán graph T r (n).

Many different proofs of Turán's theorem have been found in the literature; see [1, pp. 269-273] and [4, pp. 294-301] for more details. Furthermore, there are various extensions and generalizations on Turán's result; see, e.g., [5, 7] . In the language of extremal number, Turán's theorem can be stated as ex(n, K r+1 ) = e(T r (n)).

Moreover, we can easily see that (1 − 1 r ) n 2 2 − r 8 ≤ e(T r (n)) ≤ (1 − 1 r ) n 2 2 and e(T r (n)) = (1 − 1 r ) n 2 2 for every integer r ≤ 7. Thus Theorem 1.2 implies the explicit numerical bound e(G) ≤ (1 − 1 r ) n 2 2 for every n-vertex K r+1 -free graph G. This bound is more concise and called the weak version of Turán's theorem. The problem of determining ex(n, F ) is usually referred to as the Turán-type extremal graph problem. It is a cornerstone of extremal graph theory to understand ex(n, F ) for various graphs F ; see [18, 44] for comprehensive surveys.

Let G be a simple graph on n vertices. The adjacency matrix of G is defined as A(G) = [a ij ] ∈ R n×n where a ij = 1 if two vertices v i and v j are adjacent in G, and a ij = 0 otherwise. We say that G has eigenvalues λ 1 , λ 2 , . . . , λ n if these values are eigenvalues of the adjacency matrix A(G). Since A(G) is a symmetric real matrix, we write λ 1 , λ 2 , . . . , λ n for the eigenvalues of G in decreasing order. Let λ(G) be the maximum value in absolute among all eigenvalues of G, which is known as the spectral radius of graph G. Since the adjacency matrix A(G) is nonnegative, the Perron-Frobenius theorem (see, e.g., [56, p. 120-126] ) implies that the spectral radius of a graph G is actually the largest eigenvalue of G and it corresponds to a nonnegative eigenvector. Moreover, if G is further connected, then A(G) is an irreducible nonnegative matrix, λ(G) is an eigenvalue with multiplicity one and there exists an entry-wise positive eigenvector corresponding to λ(G).

The classical extremal graph problems usually study the maximum or minimum number of edges that the extremal graphs can have. Correspondingly, the extremal spectral problems are well-studied in the literature. The spectral Turán function ex λ (n, F ) is define to be the largest spectral radius (eigenvalue) of the adjacency matrix in an F -free n-vertex graph, that is, ex λ (n, F ) := max λ(G) : |G| = n and F G .

In 1970, Nosal [42] determined the largest spectral radius of a triangle-free graph in terms of the number of edges, which also provided the spectral version of Mantel's theorem. Note that when we consider a graph with given number of edges, we shall ignore the possible isolated vertices if there are no confusions. Theorem 1.3 (Nosal, 1970) . Let G be a graph on n vertices with m edges. If G is triangle-free, then

equality holds if and only if G is a complete bipartite graph. Moreover, we have Over the past few years, various extensions and generalizations on Nosal's theorem have been obtained in the literature; see, e.g., [36, 37, 38, 51] for extensions of K r+1free graphs, [33, 54, 55, 31] for extensions of graphs with given size. In addition, many spectral extremal problems are also obtained recently; see [11, 12] for the friendship graph and the odd wheel, [29, 13] for intersecting odd cycles and cliques, [50] for a recent conjecture. We recommend the surveys [39, 10, 28] for interested readers. The eigenvalues of the adjacency matrix sometimes can give some information about the structure of a graph. There is a rich history on the study of bounding the eigenvalues of a graph in terms of various parameters; see [6] for spectral radius and cliques, [46, 32] for eigenvalues of outerplanar and planar graphs, and [47] for the Colin de Verdière parameter, excluded minors, and the spectral radius.

In 1986, Wilf [51] provided the first result regarding the spectral version of Turán's theorem and proved that for every n-vertex K r+1 -free graph G, we have

In 2002, Nikiforov [36] proved that for every m-edge K r+1 -free graph G,

It is worth mentioning that both (3) and (4) In 2007, Nikiforov [37] showed a spectral version of the Turán theorem. Theorem 1.4 (Nikiforov, 2007) . Let G be a graph on n vertices. If G is K r+1 -free, then λ(G) ≤ λ(T r (n)), equality holds if and only if G is the r-partite Turán graph T r (n).

In other words, Nikiforov's result implies that ex λ (n, K r+1 ) = λ(T r (n)).

By calculation, we can obtain that A natural question is the following: what is the relation between the spectral Turán theorem and the edge Turán theorem? Does the spectral bound imply the edge bound of Turán's theorem? This question was also proposed and answered in [21, 38] . The answer is positive. It is well-known that e(G) ≤ n 2 λ(G), with equality if and only if G is regular. Although the Turán graph T r (n) is sometimes not regular, but it is nearly regular. Upon calculation, it follows that e(T r (n)) = n 2 λ(T r (n)) . With the help of this observation, the spectral Turán theorem implies that

Thus the spectral Turán Theorem 1.4 implies the classical Turán Theorem 1.2. To some extent, this interesting implication has shed new lights on the study of spectral extremal graph theory.

Recently, Lin, Ning and Wu [33, Theorem 1.4 ] proved a generalization of Nosal's theorem for non-bipartite triangle-free graphs (Theorem 3.2). In this paper, we shall extend the result of Lin, Ning and Wu to non-r-partite K r+1 -free graphs. Our result is also a refinement on Theorem 1.4 in the sense of stability result (Theorem 4.3). The motivation is inspired by the works [21, 37, 23, 24] , and it uses mainly the spectral extension of the Zykov symmetrization [57] . This article is organized as follows. In Section 2, we shall give an alternative proof of the spectral Turán Theorem 1.4. To make the proof of Theorem 4.3 more transparent, we will present a different proof of the result of Lin, Ning and Wu [33] in Section 3. In Section 4, we shall show the detailed proof of our main result (Theorem 4.3). In Section 5, we shall discuss the spectral extremal problem in terms of the p-spectral radius. Section 6 contains some possible open problems, including the spectral extremal problems for F -free graphs with the chromatic number χ(G) ≥ t, the problems in terms of the signless Laplacian spectral radius, and the A α -spectral radius of a graph.

2 Alternative proof of Theorem 1.4

The proof of Nikiforov [37] for Theorem 1.4 is more algebraic and based on the characteristic polynomial of the complete r-partite graph. Moreover, his proof also relies on a theorem relating the spectral radius and the number of cliques [36] , as well as an old theorem of Zykov [57] (also proved independently by Erdős [15] ), which is now known as the clique version of Turán's theorem.

In addition, the proof of Guiduli [21, pp. 58-61] for the spectral Turán theorem is completely different from that of Nikiforov. The main idea of Guiduli's proof reduces the problem of bounding the largest spectral radius among K r+1 -free graphs to complete r-partite graphs by applying a spectral extension of Erdős' degree majorization algorithm [16] . Then one can show further that the balanced complete r-partite graph attains the maximum spectral radius among all complete r-partite graphs; see, e.g., [23, 24] for more spectral applications, and [19, 3] for related topics.

In this section, we shall provide an alternative proof of Theorem 1.4. The proof is motivated by the papers [21, 23, 24] , and it is based on a spectral extension of the Zykov symmetrization [57] , which is becoming a powerful tool for extremal graph problems; see, e.g., [20] for a recent application on the minimum number of triangular edges.

For x = (x 1 , x 2 , . . . , x n ) T ∈ R n , we denote x 2 = ( n i=1 |x i | 2 ) 1/2 . Since the adjacency matrix A(G) is real and symmetric, the Rayleigh formula gives

We denote |x| = (|x 1 |, |x 2 |, . . . , |x n |) T . Suppose that x ∈ R n is an optimal vector, i.e., x 2 = 1 and λ(G) = x T A(G)x, then so is |x|. Thus there is always a non-negative unit vector x ∈ R n ≥0 such that λ(G) = x T A(G)x. Given a vector x, we know from Rayleigh's formula (or Lagrange's multiplier method) that x is an optimal vector if and only if x is a unit eigenvector corresponding to λ(G). Namely, for every v ∈ V (G), we have

This equation implies that if G is connected, then every nonnegative optimal vector is entry-wise positive. Indeed, otherwise, if x v = 0 for some v ∈ V (G), then we get x u = 0 for every u ∈ N (v). Similarly, we get x w = 0 for every w ∈ N (u). The connectivity of G leads to x w = 0 for every w ∈ V (G), and so x is a zero vector, which is a contradiction. Thus there exists an entry-wise positive eigenvector x ∈ R n >0 corresponding to λ(G) whenever G is a connected graph. This fact will be frequently used throughout the paper.

The following Lemma was proved by Feng, Li and Zhang in [17, Theorem 2.1] by using the characteristic polynomial of a complete multipartite graph; see, e.g., [25, Theorem 2] for an alternative proof and more extensions. Proof of Theorem 1.4. Let G be a K r+1 -free graph on n vertices with maximum value of the spectral radius and V (G) = {1, 2, . . . , n}. Firstly, we show that G is a connected graph. Otherwise, adding a new edge between a component attaining the spectral radius of G and any other component will strictly increase the spectral radius of G, and it does not create a copy of K r+1 . Since G is connected, we take x ∈ R n >0 as a unit positive eigenvector of λ(G). Hence, we have

Our goal is to show that G is the Turán graph T r (n). First of all, we will prove that G must be a complete t-partite graph for some integer t. Since G is K r+1 -free, this implies 2 ≤ t ≤ r. Observe that G attains the maximum spectral radius, Lemma 2.1 implies that G is further balanced, i.e., G is the t-partite Turán graph T t (n). Note that λ(T t (n)) ≤ λ(T r (n)). The maximality gives t = r and G = T r (n).

We assume on the contrary that G is not complete t-partite for every t ∈ [2, r], so there are three vertices u, v, w ∈ V (G) such that vu / ∈ E(G) and uw / ∈ E(G) while vw ∈ E(G). (This reveals that the non-edge relation between vertices is not an equivalent binary relation, as it does not satisfy the transitivity.) Throughout the paper, we denote by s G (v, x) the sum of weights of vertices in N G (v). Namely, 

We may assume that s G (u, x) < s G (v, x). Then we duplicate the vertex v, that is, we create a new vertex v which has exactly the same neighbors as v, but vv is not an edge, and we delete the vertex u and its incident edges; see the left graph in Figure 1 . Moreover, we distribute the value x u to the new vertex v , and keep the other coordinates of x unchanged. It is not hard to verify that the new graph G has still no copy of K r+1 and

where we used the positivity of vector x. This contradicts with the choice of G.

We copy the vertex u twice and delete vertices v and w with their incident edges; see the right graph in Figure 1 . Similarly, we distribute the value x v to the new vertex u , and x w to the new vertex u , and keep the other coordinates of x unchanged. Moreover, the new graph G contains no copy of K r+1 and

So we get a contradiction again.

We conclude here that the spectral version of Zykov's symmetrization starts with a K r+1 -free graph G with vertex set V = {1, 2, . . . , n}, and at each step takes two

, and deleting all edges incident to v j , and adding new edges between vertex v j and the neighborhood N

The spectral version of Zykov's symmetrization does not increase the size of the largest clique and does not decrease the spectral radius 1 . When the process terminates, it yields a complete multipartite graph with at most r vertex parts. Otherwise, there are three vertices u, v, w ∈ V (G) such that vu / ∈ E(G) and uw / ∈ E(G) but vw ∈ E(G). Applying the same case analysis as in Theorem 1.4, we will get a new graph with larger spectral radius, which is a contradiction.

We illustrate the difference of the spectral extension between the Erdős degree majorization algorithm and the Zykov symmetrization. Recall that the spectral version of the Erdős degree majorization algorithm asks us to choose a vertex v ∈ V (G) with maximum value of s G (v, x) among all vertices of G, and we remove all edges incident to vertices of V (G) \ (N G (v) ∪ {v}), and then add all edges between N G (v) and V (G)\N G (v). We observe that this operation makes each vertex of V (G)\(N G (v)∪{v} being a copy of the vertex v. Since G is K r+1 -free, we see that the subgraph of G induced by N G (v) is K r -free. We denote by

, and then add all edges between N V c 1 (u) and V c 1 \ N V c 1 (u). By using this operation repeatedly, we get a complete r-partite graph H on the same vertex set V (G). Furthermore, one can verify that the majorization inequality s G (v, x) ≤ s H (v, x) holds for every vertex v ∈ V (G); see, e.g., [21, 23, 24] .

The spectral extensions of the Erdős majorization algorithm and the Zykov symmetrization share some similarities. For example, these two operations ask us to compare the sum of weights of neighbors, and turn a K r+1 -free graph into a complete r-partite. Importantly, these two operations do not create a copy of K r+1 and do not decrease the value of spectral radius. The only difference between them is that one step of the Erdős operation will change many vertices and its incident edges, while one step of the Zykov operation will only change two vertices and its incident edges. This subtle difference will bring great convenience in later Sections 3 and 4. As a matter of fact, at each step of the Erdős operation, there are many times of actions of the Zykov operation. In other words, each step of the Erdős operation can be decomposed as a series of the Zykov operation.

Mantel's theorem has many interesting applications and miscellaneous generalizations in the literature; see, e.g., [4, 5, 7, 44] and references therein. In particular, Mantel's Theorem 1.1 was refined in the sense of the following stability form. It is said that this stability result attributes to Erdős; see [8, Page 306, Exercise 12.2.7]. The bound in Theorem 3.1 is best possible and the extremal graph is not unique. To show that the bound is sharp for all integers n, we take two vertex sets X and Y with |X| = n 2 and |Y | = n 2 . We take two vertices u, v ∈ Y and join them, then we put every edge between X and Y \ {u, v}. We partition X into two parts X 1 and X 2 arbitrarily (this shows that the extremal graph is not unique), then we connect u to every vertex in X 1 , and v to every vertex in X 2 ; see Figure 2 . This yields a graph G which contains no triangle and e(G) = n 2 4 − n 2 + 1 = (n−1) 2 4 + 1. Note that G has a 5-cycle, so G is not bipartite. ; see [34, 27] for a recent extension on graphs without short odd cycles, and [30] for more stability theorems on spectral graph problems.

In this section, we shall provide an alternative proof of Theorem 3.2. One of the key ideas in the proof is to use the spectral Zykov symmetrization, which provides great convenience to yield a clearly approximate structure of the required extremal graph. Moreover, the ideas in this proof can benefit us to extend Theorem 3.2 to K r+1free non-r-partite graphs, which will be discussed in Section 4. Before starting the proof, we include the following lemma, which is a direct consequence by computations; see, e.g., [33, Appendix A]. Proof. We denote by SK a,b the graph obtained from K a,b by subdividing an edge. Let s, t be two positive integers with t ≥ s ≥ 1. It suffices to show that

By computation, the spectral radius of SK a,b is the largest root of

Hence λ(SK s+2,t+2 ) is the largest root of

Similarly, λ(SK s+1,t+3 ) is the largest root of F s+1,t+3 (x). Note that

Therefore, we obtain λ(SK s+1,t+3 ) < λ(SK s+2,t+2 ).

Now we are ready to show our proof of Theorem 3.2. For two non-adjacent vertices u, v ∈ V (G), we denote the Zykov symmetrization Z u,v (G) to be the graph obtained from G by replacing u with a twin of v, that is, deleting all edges incident to vertex u, and then adding new edges from u to N G (v). We can verify that the Zykov symmetrization does not increase both the clique number ω(G) and the chromatic number

Apparently, the spectral Zykov symmetrization does not make triangles. More importantly, it will increase strictly the spectral radius, since

, then we can apply either Z u,v or Z v,u , which leads to N (u) = N (v) after making the spectral Zykov symmetrization, and while the operation will keep the spectral radius λ(G) increasing strictly. Indeed, we can easily see that

We claim that λ(Z u,v (G)) > λ(G). Assume on the contrary that λ(Z u,v (G)) = λ(G), then the inequality in above become an equality, thus x is an eigenvector

, which contradicts with our assumption. It is worth emphasizing that the positivity of x is necessary in above discussions. Roughly speaking, applying the spectral Zykov symmetrization will make the K r+1free graph more regular in some sense according to the weights of the eigenvector.

Proof of Theorem 3.2. Let G be a non-bipartite triangle-free graph on n vertices with the largest spectral radius. Our goal is to show that G = SK n−1 2 , n−1 2 . Clearly, we know that G is connected. Otherwise, any addition of an edge between a component with the maximum spectral radius and any other component will strictly increase the spectral radius. Since G is connected, there exists a positive unit eigenvector corresponding to λ(G), and then we denote such a vector by x = (x 1 , . . . , x n ) T , where x i > 0 for every i. Since G is triangle-free, we apply repeatedly the spectral Zykov symmetrization for every pair of non-adjacent vertices until it becomes a bipartite graph. Without loss of generality, we may assume that G is trianglefree and non-bipartite, while Z u,v (G) is bipartite. We next are going to show that Note that each vertex in A has the same neighborhood, we know that the coordinates {x v : v ∈ A} are all equal. This property holds similarly for vertices in B, C and D respectively. Thus, we write x a for the value of the entries of x in vertex set A. And x b , x c and x d are defined similarly.

The remaining steps of our proof are outlined as follows.

$ If |A|x a ≥ |B|x b , then we delete |C| − 1 vertices in C with its incident edges, and add |C| − 1 vertices to D and connect these vertices to A ∪ {u}. We keep the weight of these new vertices being x c and denote the new graph by G . We can verify that λ

In fact, we can further prove that λ(G ) > λ(G). Otherwise, if λ(G ) = λ(G), then x is the Perron vector of G , that is, A(G ) = λ(G )x = λ(G)x. Taking any vertex z ∈ A, we observe that

and then λ(G ) > λ(G), which is a contradiction.

) If |A|x a < |B|x b , then we can delete |D| − 1 vertices from D with its incident edges, and add |D|−1 vertices to C and join these vertices to B ∪{u}. Similarly, we can show that this process will increase the spectral radius strictly. From the above discussion, we can always remove the vertices to force either |C| = 1 or |D| = 1. Without loss of generality, we may assume that |C| = 1 and C = {c}.

( If x u ≥ x c , then we remove |B| − 1 vertices from B with its incident edges, and add |B| − 1 vertices to D and join these vertices to A ∪ {u}. We keep the weight of these new vertices being x b and denote the new graph by G * . Then λ( From our discussion above, we know that if G is an n-vertex triangle-free nonbipartite graph and attains the maximum spectral radius, then G is a subdivision of a complete bipartite by subdividing exactly one edge. Lemma 3.3 implies that G is a subdivision of a balanced complete bipartite graph on n − 1 vertices.

In 1981, Brouwer [9] proved the following improvement on Turán's Theorem 1.2.

Theorem 4.1 (Brouwer, 1981) . Let n ≥ 2r + 1 be an integer and G be an n-vertex graph. If G is K r+1 -free and G is not r-partite, then e(G) ≤ e(T r (n)) − n r + 1.

Theorem 4.1 was also independently studied in many references, e.g., [2, 22, 26, 49] . Similar with that of Theorem 3.1, the bound of Theorem 4.1 is sharp and there are many extremal graphs attaining this bound.

We would like to illustrate the reason why we are interested in the study of the family of non-r-partite graphs. On the one hand, the Erdős degree majorization algorithm [16] or [4, pp. 295-296] implies that if G is an n-vertex K r+1 -free graph, then there exists an r-partite graph H on the same vertex set V (G) such that d G (v) ≤ d H (v) for every vertex v. Consequently, we get e(G) ≤ e(H) ≤ e(T r (n)). Hence it is meaningful to determine the family of graphs attaining the second largest value of the extremal function. This problem is also called the stability problem. On the other hand, there are various ways to study the extremal graph problems under some reasonable constraints. For example, the condition of non-r-partite graph is equivalent to saying the chromatic number χ(G) ≥ r + 1. Moreover, we can also consider the extremal problem under the restriction α(G) ≤ f (n) for a given function f (n), where α(G) is the independence number of G. This is the well-known Ramsey-Turán problem; see [45] for a comprehensive survey.

The proof of Theorem 3.2 stated in Section 3 can bring us more effective treatment for the extremal spectral problem when K r+1 is a forbidden subgraph. In what follows, we shall extend Lin-Ning-Wu's Theorem 3.2. Our result is also a spectral version of Brouwer's Theorem 4.1.

Recall that T r (n) is the n-vertex r-partite Turán graph in which the parts T 1 , T 2 , . . . , T r have sizes t 1 , t 2 , . . . , t r respectively. We may assume that n r = t 1 ≤ t 2 ≤ · · · ≤ t r = n r . Now, we are going to define a new graph obtained from T r (n). Firstly, we choose two vertex parts T 1 and T r . Secondly, we add a new edge into T r , denote by uw, and then remove all edges between T 1 and {u, w}. Finally, we connect u to a vertex v ∈ T 1 , and connect w to the remaining vertices of T 1 . The resulting graph is denoted by Y r (n); see Figure 5 . Clearly, Y r (n) is one of the extremal graphs of Brouwer's theorem. Let G be an n-vertex graph obtained from K b 1 ,b 2 ,...,br by adding a new vertex u and choosing v ∈ B 1 , w ∈ B 2 , and removing the edge vw, and adding the edges uv, uw and ut for every t ∈ ∪ r i=3 B i . Then λ(G) ≤ λ(Y r (n)).

Moreover, the equality holds if and only if G = Y r (n).

Next, we illustrate the construction of Y r (n) in another way. Let T r (n − 1) be the r-partite Turán graph on n − 1 vertices whose parts S 1 , S 2 , . . ., S r have sizes s 1 , s 2 , . . . , s r such that n−1 r = s 1 ≤ s 2 ≤ · · · ≤ s r = n−1 r . Note that Y r (n) can also be obtained from T r (n − 1) by adding a new vertex u, and choosing two vertices v ∈ S 1 and w ∈ S 2 , and deleting the edge vw, and adding the edges uv, uw and ut for every vertex t ∈ ∪ r i=3 S i . Hence Lemma 4.2 states that G attains the maximum spectral radius when its part sizes b 1 , b 2 , . . . , b r are as equal as possible and the two special vertices v, w are located in the smallest two parts respectively. We know that λ(G) is the largest root of the characteristic polynomial P G (x) = det(xI n − A(G)). It is operable to compute λ(G) exactly for some small integers r by using computers, while it seems complicated for large r.

Proof of Lemma 4.2. Let G be a graph satisfying the requirement of Lemma 4.2 and G has the maximum spectral radius. We will show that G = Y r (n). Since G is connected, there exists a positive unit eigenvector x ∈ R n corresponding to λ(G). Assume on the contrary that G is not isomorphic to Y r (n). In other words, there are two parts B i and B j such that |b i −b j | ≥ 2, or b i ≤ b j −1 for some i ∈ {3, 4, . . . , n} and j ∈ {1, 2}. By the symmetry, there are four cases listed below.

Case 1. First and foremost, we shall consider case (i) that b i ≤ b j − 2 for some i, j ∈ {3, . . . , r}. The treatment for this case has its root in [25] . If b i +b j = 2b for some integer b, then we will balance the number of vertices of parts B i and B j . Namely, we define a new graph G obtained from G by deleting all edges between B i and B j , and then we move some vertices from B j to B i such that the resulting sets, say B i , B j , have size b, and then we add all edges between B i and B j . In this process, we keep the other edges unchanged. We define a vector y ∈ R n such that y s = (

Moreover, the weighted power-mean inequality gives

which contradicts with the choice of G. If b i + b j = 2b + 1 for some integer b, then we move similarly some vertices from B j to B i such that the resulting sets B i , B j satisfying |B i | = b and |B j | = b + 1. We construct a vector y ∈ R n satisfying y s = ( b i x 2 i +b j x 2 j 2b+1 ) 1/2 for every vertex s ∈ B i ∪ B j , and y t = x t for every t ∈ V (G ) \ (B i ∪ B j ). Similarly, we get

We are going to show that

For the first inequality, by applying AM-GM inequality, we get

Then it is sufficient to prove that 2b(b + 1)

The desired inequality holds. For the second one, the weighted power-mean inequality yields (2b + 1)y s = (2b + 1)

This case also contradicts with the choice of G.

For the remaining three cases, we will show our proof by considering the characteristic polynomial of G and then applying induction on integer r.

We define a graph G obtained from G by deleting a vertex of B − 2 , and adding a copy of a vertex of B − 1 . This makes the two parts B − 1 , B − 2 more balanced. Our goal is to prove that λ(G) < λ(G ), which contradicts with the maximality of G. Let x v , x w and x u be the weights of vertices v, w and u respectively. We denote by x − 1 and x − 2 the weights of vertices of B − 1 and B − 2 respectively. The eigen-equation

Thus λ(G) is the largest eigenvalue of the matrix A r corresponding to eigenvector

is defined as the following.

For notational convenience, we denote

For every r ≥ 2, the characteristic polynomial of A r is denoted by

In particular, the polynomial F b 1 ,b 2 (x) is the same as that in Lemma 3.3. By expanding the last column of det(xI r+3 − A r ), we get the following recurrence relations:

and for every integer r ≥ 4,

where F b 1 ,b 2 (x) and R b 1 ,b 2 (x) are computed as below:

Upon computations, we obtain

Combining with equation (5), we obtain

Next we prove by induction that for every r ≥ 3 and x ≥ 2,

Firstly, the base case r = 3 was verified in the above. For r ≥ 4, we get from (6) that

where the last inequality holds by applying inductive hypothesis on the case r − 1 and invoking the fact

..,br (x), this implies λ(G) < λ(G ). Case 3. Thirdly, we consider case (iii) that b 1 ≤ b i − 2 for some i ∈ {3, . . . , r}. We may assume by symmetry that b 1 ≤ b 3 − 2. Our treatment in this case is similar with that of case (ii). Let G * be the graph obtained from G by deleting a vertex of B 3 with its incident edges, and add a new vertex to B − 1 and connect this new vertex to all remaining vertices of B 3 and all vertices of B 2 ∪ B 4 ∪ · · · ∪ B r . We will prove that λ(G) < λ(G * ). By case (ii), we may assume that |b 1 

..,br (x). First of all, we will show that

and then by applying induction, we will prove that for each r ≥ 4,

Indeed, we next verify inequalities (8) and (9) for the case r = 4 only, since the inductive steps are the same as that of case (ii) with slight differences. By computation, we obtain

Combining these two equations with (5), we get

Combining

. This completes the proof of (8). We now consider (9) in the case r = 4. Note that b 1 − b 3 + 2 ≤ 0 and

which together with (6) and the case r = 3 yields

Since the complete r-partite K t,t,...,t is a subgraph of G, we know that λ(G) ≥ λ(K t,t,...,t ) = (r − 1)t. Thus, we can similarly get that

..,br (λ(G)) = 0, which yields λ(G) < λ(G * ), which contradicts with the choice of G. Case 4. Finally, we consider the case (iv) that b i ≤ b 1 − 1 for some i ≥ 3. We may assume that b 3 ≤ b 1 − 1. This case can similarly be completed by applying a similar argument of case (iii). Let G * be the graph obtained from G by removing a vertex of B − 1 with its incident edges, and adding a copy of a vertex of B 3 . In what follows, we will show that

and then we prove by induction that for every r ≥ 4,

By computation, we obtain that

Combining with the recurrence equation (5), we get

. This completes the proof of (10). Next we will prove (11) for the case r = 4 only, since the inductive steps are similar with that of case (ii) and (iii). By computation, we have

which together with (6) and the case r = 3 gives

..,br (λ(G)) = 0 and λ(G * ) is the largest root of F b 1 −1,b 2 ,b 3 +1,b 4 ,...,br (x), we know that λ(G) < λ(G * ), which contradicts with the choice of G. In summary, we complete the proof of all possible cases.

Remark. It seems possible to prove the last three cases by using a weight-balanced argument similar with that of the first case. Nevertheless, it is inevitable that a great deal of tedious calculations are required in the proof of these cases. Moreover, applying the recursive technique of determinants in the proof of Lemma 4.2, one can compute the characteristic polynomial of the adjacency matrix and signless Laplacian matrix of the n-vertex complete r-partite graph K t 1 ,...,tr . More precisely,

It has its own interests to compute the eigenvalues of complete multipartite graphs; see, e.g., [14, 53, 43, 52] for different proofs and related results.

We next show our main result in this paper. Theorem 4.3. Let G be an n-vertex graph. If G is K r+1 -free and G is not r-partite, then λ(G) ≤ λ(Y r (n)).

Moreover, the equality holds if and only if G = Y r (n). Theorem 4.3 is a refinement of the spectral Turán Theorem 1.4 and also it is an extension of Theorem 3.2. Our proof is mainly based on Zykov's symmetrization.

Proof. First of all, we assume that G is a K r+1 -free non-r-partite graph with maximum value of spectral radius. Our goal is to prove that G = Y r (n). Clearly, we know that G must be a connected graph. Let x ∈ R n >0 be a positive unit eigenvector of λ(G). Claim 4.1. There exists a vertex u ∈ V (G) such that G \ {u} is r-partite.

Proof of Claim 4.1. Recall that for two non-adjacent vertices u, v ∈ V (G), the spectral Zykov symmetrization Z u,v (G) is defined as the graph obtained from G by removing all edges incident to vertex u and then adding new edges from u to N G (v). We can verify that the spectral Zykov symmetrization does not increase the clique number and the chromatic number. Recall that

, then we can apply either Z u,v or Z v,u , which leads to N (u) = N (v) after making the spectral Zykov symmetrization. Obviously, the spectral Zykov symmetrization does not create a copy of K r+1 . More significantly, it will increase the spectral radius strictly, since x is entry-wise positive.

The proof of Claim 4.1 is based on the spectral Zykov symmetrization stated in above. Since G is K r+1 -free, we can repeatedly apply the Zykov symmetrization on every pair of non-adjacent vertices until G becomes an r-partite graph. Without loss of generality, we may assume that G is K r+1 -free and G is not r-partite, while Z u,v (G) is r-partite. Thus G \ {u} is r-partite, and we assume that V (G) \ {u} = V 1 ∪ V 2 ∪ · · · ∪ V r , where V 1 , V 2 , . . . , V r are pairwise disjoint and r i=1 |V i | = n − 1. We denote A i = N (u) ∩ V i for every i ∈ [r] := {1, . . . , r}. Note that G has maximum spectral radius among all K r+1 -free non-r-partite graphs. Then for each i ∈ [r], every vertex of V i \ A i is adjacent to every vertex of V j for every j ∈ [r] and j = i. We remark here that the difference between the K r+1 -free case (Theorem 4.3) and the triangle-free case (Theorem 3.2) is that there may exist some edges between the pair of sets A i and A j , which makes the problem seems more difficult. Proof of Claim 4.2. Let G[A 1 , A 2 , . . . , A r ] be the subgraph of G induced by the vertex sets A 1 , A 2 , . . . , A r . Claim 4.2 is equivalent to say that G[A 1 ∪ A 2 , A 3 , . . . , A r ] forms a complete (r − 1)-partite subgraph in G. Since G is K r+1 -free, we know that the subgraph G[A 1 , A 2 , . . . , A r ] is a K r -free subgraph in G.

First of all, we choose a vertex v 1 ∈ A 1 such that s G (v 1 , x) is maximum among all vertices of A 1 , then we apply the Zykov operation Z u,v 1 on G for every u ∈ A 1 \ {v 1 }. These operations will make all vertices of A 1 being equivalent, that is, every pair of vertices in A 1 has the same neighbors. Secondly, we choose a vertex v 2 ∈ A 2 such that s G (v 2 , x) is maximum over all vertices of A 2 , and then we apply similarly the Zykov operation Z u,v 2 on G for every u ∈ A 2 \ {v 2 }. Note that all vertices in A 1 have the same neighbors. After doing Zykov's operations on vertices of A 2 , we claim that the induced subgraph G[A 1 , A 2 ] is either a complete bipartite graph or an empty graph. Indeed, if v 2 ∈ ∩ v∈A 1 N (v), then the operations Z u,v 2 for all u ∈ A 2 \ {v 2 } will lead to a complete bipartite graph between A 1 and A 2 . If v 2 ∈ ∩ v∈A 1 N (v), then v 2 is not adjacent to all vertices of A 1 , and so is u for every u ∈ A 2 \ {v 2 }, which yields that G[A 1 , A 2 ] is an empty graph. Moreover, by applying the similar operations on A 3 , A 4 , . . . , A r , we can obtain that for every i, j ∈ [r] with i = j, the induced bipartite subgraph G[A i , A j ] is either complete bipartite or empty. Since G[A 1 , A 2 , . . . , A r ] is K r -free and G attains the maximum spectral radius, we know that there is exactly one pair {i, j} ⊆ [r] such that G[A i , A j ] is an empty graph.

We may assume that {i, j} = {1, 2} for convenience. In what follows, we intend to enlarge A i to the whole set V i for every i ∈ {3, 4, . . . , r}. Observe that every vertex of V i \ A i is adjacent to every vertex of V j for every j ∈ [r] with j = i, and adding all edges between {u} to V i \ A i does not create a copy of K r+1 in G, and does not decrease the spectral radius of G. From this observation, we know that u is adjacent to every vertex of V i for each i ∈ {3, 4, . . . , r}; see (a) in Figure 6 . Figure 6 . Note that there is no edge between C and D, since G does not contain K r+1 as a subgraph. In the remaining of our proof, we will prove by two steps that both C and D are single vertex sets. Proof of Claim 4.3. The treatment is similar with that of our proof of Theorem 3.2.

If v∈A x v ≥ v∈B x v , then we choose |C| − 1 vertices of C and delete its incident edges only in B, then we move these |C|−1 vertices into D and connect these vertices to A. In this process, the edges between these |C| − 1 vertices and V 3 ∪ · · · ∪ V r are unchanged. We write G for the resulting graph. Using the similar computation as in Section 3, we can verify that λ(G ) > λ(G).

If v∈A x v < v∈B x v , then we can choose |D| − 1 vertices of D and delete its incident edges only in A, and then move these |D| − 1 vertices into C and join these vertices to B. This process will increase strictly the spectral radius.

From the above case analysis, we can always remove the vertices of G to force either |C| = 1 or |D| = 1. Without loss of generality, we may assume that |C| = 1 and denote C = {c}; see (b) in Figure 6 . Proof of Claim 4.4. If x u < x c , then we choose |D| − 1 vertices of D and delete its incident edges to vertex u, then we move these |D| − 1 vertices into B and join these these vertices to c, and keeping the other edges unchanged, we denote the new graph by G . Then we can similarly get λ(G ) > λ(G). In the graph G , we have |D| = 1 and write D = {d}. Thus G is the graph obtained from a complete r-partite graph K t 1 ,t 2 ,...,tr , where r i=1 t i = n − 1, by adding a new vertex u and then joining u to a vertex c ∈ V 1 , and joining u to a vertex d ∈ V 2 , and joining u to all vertices of V 3 ∪ · · · ∪ V r , and finally removing the edge cd ∈ E(K t 1 ,t 2 ,...,tr ).

If x u ≥ x c , then we choose |B| − 1 vertices of B and delete its incident edges to vertex c, then we move these |B| − 1 vertices into D and join these vertices to vertex u. We denote the new graph by G * . Then λ(G * ) > λ(G). Thus we conclude in the new graph G * that B is a single vertex, say B = {b}; see (c) in Figure 6 . In what follows, we will exchange the position of u and c. Note that c ∈ V 1 is adjacent to a vertex b ∈ V 1 and all vertices of V 3 ∪ · · · ∪ V r . Now, we move vertex c outside of V 1 and put vertex u into V 1 . Thus the new center c is adjacent to a vertex u ∈ V 1 , a vertex b ∈ V 2 and all vertices of V 3 ∪ · · · ∪ V r . Note that bu / ∈ E(G * ). Hence G * has the same structure as the previous case, and then we may assume that |D| = 1.

From the above discussion, we know that G is isomorphic to the graph defined as in Lemma 4.2. By applying Lemma 4.2, we know that λ(G) ≤ λ(Y r (n)). Moreover, the equality holds if and only if G = Y r (n). This completes the proof.

Recall that the spectral radius of a graph is defined as the largest eigenvalue of its adjacency matrix. By Rayleigh's theorem, we know that it is also equal to the maximum value of x T A(G)x = 2 {i,j}∈E(G) x i x j over all x ∈ R n with |x 1 | 2 + · · · + |x n | 2 = 1. The definition of the spectral radius was recently extended to the p-spectral radius. We denote the p-norm of x by x p = (|x 1 | p + · · · + |x n | p ) 1/p . For every real number p ≥ 1, the p-spectral radius of G is defined as

{i,j}∈E(G)

x i x j .

We remark that λ (p) (G) is a versatile parameter. Indeed, λ (1) (G) is known as the Lagrangian function of G, λ (2) (G) is the spectral radius of its adjacency matrix, and

which can be guaranteed by the following inequality

To some extent, the p-spectral radius can be viewed as a unified extension of the classical spectral radius and the size of a graph. In addition, it is worth mentioning that if 1 ≤ q ≤ p, then λ (p) (G)n 2/p ≤ λ (q) (G)n 2/q and (λ (p) (G)/2e(G)) p ≤ (λ (q) (G)/2e(G)) q ; see [40, Propositions 2.13 and 2.14] for more details.

As commented by Kang and Nikiforov in [24, p. 3] , linear-algebraic methods are irrelevant for the study of λ (p) (G) in general, and in fact no efficient methods are known for it. Thus the study of λ (p) (G) for p = 2 is far more complicated than the classical spectral radius.

The extremal function for p-spectral radius is given as ex (p) λ (n, F ) := max{λ (p) (G) : |G| = n and G is F -free}.

To some extent, the proof of results on the p-spectral radius shares some similarities with the usual spectral radius when p > 1; see [40, 24, 25] for extremal problems for the p-spectral radius. In 2014, Kang and Nikiforov [24] extended the Turán theorem to the p-spectral version for p > 1. They proved that ex (p) λ (n, K r+1 ) = λ (p) (T r (n)).

Theorem 5.1 (Kang-Nikiforov, 2014 ). If G is a K r+1 -free graph on n vertices, then for every p > 1,

equality holds if and only if G is the n-vertex Turán graph T r (n).

Remark. We remark that a theorem of Motzkin and Straus states that Theorem 5.1 is also valid for the case p = 1 except for the extremal graphs attaining the equality.

Keeping (13) in mind, we can see that Theorem 5.1 is a unified extension of both Turán's Theorem 1.2 and Spectral Turán's Theorem 1.4 by taking p → +∞ and p = 2 respectively. We can obtain by detailed computation that λ (p) (T r (n)) = (1 + O( 1 n 2 ))2e(T r (n))n −2/p and λ (p) (T r (n)) = (1 − O( 1 n 2 )) 1 − 1

stands for a positive error term. This theorem implies λ (p) (G) ≤ 1 − 1 r n 2−(2/p) , equality holds if and only if r divides n and G = T r (n).

Recall that the proof of Theorem 4.3 relies on the Rayleigh representation of λ(G) and the existence of a positive eigenvector of λ(G). For the p-spectral radius, there is also a positive vector corresponding to λ (p) (G). Indeed, we choose G as a K r+1free graph on n vertices with maximum value of the p-spectral radius, where p > 1. Clearly, we can assume further that G is connected. A vector x ∈ R n is called a unit (optimal) eigenvector corresponding to λ (p) (G) if it satisfies n i=1 |x i | p = 1 and λ (p) (G) = 2 {i,j}∈E(G) x i x j . From the definition (12), we know that there is always a non-negative eigenvector of λ (p) (G). Moreover, since p > 1, Lagrange's multiplier method gives that for every v ∈ V (G), we have

Therefore, if G is connected and p > 1, applying (14), then a non-negative eigenvector of λ (p) (G) must be entry-wise positive. Hence there exists a positive unit optimal vector x ∈ R n >0 corresponding to λ (p) (G) such that λ (p) (G) = 2

{i,j}∈E(G)

x i x j .

By applying a similar line of the proof of Theorem 4.3, one can extend Theorem 4.3 to the p-spectral radius. We leave the details for interested readers.

Theorem 5.2. Let G be an n-vertex graph. If G does not contain K r+1 and G is not r-partite, then for every p > 1, we have λ (p) (G) ≤ λ (p) (Y r (n)).

Moreover, the equality holds if and only if G = Y r (n).

In this paper, we studied the spectral extremal graph problems for graphs with given number of vertices. By extending the Mantel theorem and Nosal theorem, we presented an alternative proof of an extension of Nikiforov for K r+1 -free graphs, and provided a different proof of a refinement of Lin, Ning and Wu for non-bipartite K 3free graphs. Furthermore, we generalized these two results to non-r-partite K r+1 -free graphs. Our result is not only a refinement on the spectral Turán theorem, but it is also a spectral version of Brouwer's theorem. In a forthcoming paper [31] , we shall present some extensions and generalizations on Nosal's theorem for graphs with given number of edges.

At the end of this paper, we shall conclude with some possible problems for interested readers. To begin with, we define an extremal function as ψ(n, F, t) := max{e(G) : F G, χ(G) ≥ t}.

Brouwer's theorem says that ψ(n, K r+1 , r + 1) = e(T r (n)) − n r + 1. Similarly, we can define the spectral extremal function as ψ λ (n, F, t) := max{λ(G) : F G, χ(G) ≥ t}.

In Theorem 4.3, we proved that ψ λ (n, K r+1 , r+1) = λ(Y r (n)). Note that the extremal graph Y r (n) has chromatic number χ(Y r (n)) = r + 1. It is possible to determine the function ψ λ (n, K r+1 , r + 2). More generally, it would be interesting to determine the functions ψ(n, F, t) and ψ λ (n, F, t) for a general graph F and an integer t. For instance, it is possible to study these extremal functions by setting F as the odd cycle C 2k+1 , the book graph B k = K 2 ∨ kK 1 , the fan graph F k = K 1 ∨ kK 2 , the wheel graph W k = K 1 ∨ C k , and a color-critical graph F .

We write q(G) for the signless Laplacian spectral radius, i.e., the largest eigenvalue of the signless Laplacian matrix Q(G) = D(G)+A(G), where D(G) = diag(d 1 , . . . , d n ) is the degree diagonal matrix and A(G) is the adjacency matrix. In 2013, He, Jin and Zhang [23, Theorem 1.3] showed some bounds for the signless Laplacian spectral radius in terms of the clique number. As a consequence, they proved the signless Laplacian spectral version of Theorem 1.2, which states that if G is a K r+1 -free graph on n vertices, then q(G) ≤ q(T r (n)), equality holds if and only if r = 2 and G = K t,n−t for some t, or r ≥ 3 and G = T r (n). This spectral extension also implies the classical edge Turán theorem. It is possible to establish analogues of the results of our paper in terms of the signless Laplacian spectral radius. For example, whether Y r (n) is the extremal graph attaining the maximum signless Laplacian spectral radius among all non-r-partite K r+1 -free graphs.

In 2017, Nikiforov [41] provided a unified extension of both the adjacency spectral radius and the signless Laplacian spectral radius. It was proposed by Nikiforov [41] to study the family of matrices A α defined for any real α ∈ [0, 1] as A α (G) = αD(G) + (1 − α)A(G).

In particular, we can see that A 0 (G) = A(G) and 2A 1/2 (G) = Q(G). Nikiforov [41, Theorem 27] presented some extremal spectral results in terms of the spectral radius of A α . It was proved that for every r ≥ 2 and every K r+1 -free graph G, if 0 ≤ α < 1 − 1 r , then λ(A α (G)) < λ(A α (T r (n))), unless G = T r (n); if α = 1 − 1 r , then λ(A α (G)) < (1 − 1 r )n, unless G is a complete r-partite graph; if 1 − 1 r < λ < 1, then λ(A α (G)) < λ(A α (S n,r−1 )), unless G = S n,r−1 , where S n,k = K k ∨ I n−k is the graph consisting of a clique on k vertices and an independent set on n − k vertices in which each vertex of the clique is adjacent to each vertex of the independent set. From this evidence, it is possible to extend the results of our paper into the A α -spectral radius in the range α ∈ [0, 1 − 1 r ) for non-r-partite K r+1 -free graphs.

Proofs from THE BOOK

On the non-(p − 1)-partite K p -free graphs

The typical structure of graphs with no large cliques

Extremal Graph Theory

Dense neighbourhoods and Turán's theorem

Cliques and the spectral radius

Large dense neighbourhoods and Turán's theorem

Graph Theory

Some Lotto Numbers From an Extension of Turán's Theorem

Some new results and problems in spectral extremal graph theory

The spectral radius of graphs with no intersecting triangles

The spectral radius of graphs with no odd wheels

Spectral extremal graphs for intersecting cliques

Eigenvalues of complete multipartite graphs

On the number of complete subgraphs contained in certain graphs

On the graph theorem of Turán

Spectral radii of graphs with given chromatic number

The history of degenerate (bipartite) extremal graph problems

A proof of the stability of extremal graphs, Simonovits' stability from Szemerédi's regularity

The minimum number of triangular edges and a symmetrization method for multiple graphs

Spectral extrema for graphs

Toft, k-saturated graphs of chromatic number at least k, Ars Combin

Sharp bounds for the signless Laplacian spectral radius in terms of clique number

Extremal problem for the p-spectral radius of graphs

The p-spectral radius of k-partite and kchromatic uniform hypergraphs

Maximum K r+1 -free graphs which are not r-partite

Adjacency eigenvalues of graphs without short odd cycles

A survey on spectral conditions for some extremal graph problems

The spectral radius of graphs with no intersecting odd cycles

New proofs of stability theorems on spectral graph problems

The maximum spectral radius of non-bipartite graphs forbidding short odd cycles

A complete solution to the Cvetković-Rowlinson conjecture

Eigenvalues and triangles in graphs

A spectral condition for odd cycles in non-bipartite graphs

Some inequalities for the largest eigenvalue of a graph

Bounds on graph eigenvalues II

More spectral bounds on the clique and independence numbers

Some new results in extremal graph theory, Surveys in Combinatorics

Analytic methods for uniform hypergraphs

Merging the A-and Q-spectral theories

Eigenvalues of graphs

A relation between the signless Laplacian spectral radius of complete multipartite graphs and majorization

Paul Erdős' influence on extremal graph theory

Combinatorics, Graph Theory

Three conjectures in extremal spectral graph theory

The Colin de Verdière parameter, excluded minors, and the spectral radius

On an extremal problem in graph theory

Strong Turán stability

On a conjecture of spectral extremal problems

Spectral bounds for the clique and indendence numbers of graphs

On the spectrum of an equitable quotient matrix and its application

Signless Laplacian spectral radii of graphs with given chromatic number

Spectral extrema of graphs with fixed size: Cycles and complete bipartite graphs

A spectral version of Mantel's theorem

Matrix Theory

On some properties of linear complexes

This work was supported by NSFC (Grant No. 11931002). This article was completed during a quarantine period due to the COVID-19 pandemic. The authors would like to express their sincere gratitude to all of the volunteers and medical staffs for their kind help and support, which makes our daily life more and more secure.