A spectral Turán theorem

Transcription

1 A spectral Turán theorem Fan Chung Abstract If all nonzero eigenalues of the (normalized) Laplacian of a graph G are close to, then G is t-turán in the sense that any subgraph of G containing no K t+ contains at most ( /t + o())e(g) edges where e(g) denotes the number of edges in G. Introduction One of the classical theorems in graph theory is Turán s Theorem which states that a graph on n ertices containing no K t+ canhaeatmost( /t + o()) ( n 2) edges. Sudako, Szabó and Vu [6] consider a generalization of Turán s Theorem. A graph G is said to be t-turán if any subgraph of G containing no K t+ hasatmost( /t+o())e(g) edges where e(g) denotes the number of edges in G. In[6],itisshown that a regular graph on n ertices with degree d is t-turán if the second largest eigenalue of its adjacency matrix λ is sufficiently small. In this paper, we consider Turán numbers for general graphs as introduced in [6]. For two gien graphs G and H, theturán number t(g, H) is defined to be t(g, H) =max{e(g ) : G is a subgraph of G containing no H}. The classical Turán number is the special case that G is a complete graph K n.turán s theorem implies t(k n,k t+ )=( t ( ) n +o()). t 2 In this paper, we will show that t(g, K t+ )=( t t +o())e(g) () as long as certain spectral bounds of G are satisfied (to be specified in Section 4). Since any t-partite subgraph of G contains no K t+, the inequality t(g, K t+ ) ( t +o())e(g) always holds. Thus, equation () implies that a maximum t-partite subgraph of G is an extremal graph haing the Uniersity of California, San Diego Research supported in part by NSF Grants DMS , ITR and ITR

2 maximum number of edges among all subgraphs of G containing no K t+. In section 4, we will show that our main theorem implies (the asymptotical ersion of) the classical Turán s Theorem as a special case. Another consequence of our main theorem is the result in [6] for d-regular graphs. Namely, if the second largest eigenalue µ of the adjacency matrix of a d-regular graph on n ertices satisfies µ d t /n t,then t(g, K t+ )=( /t + o())dn/2. This will also be proed in Section 4. In order to derie the relationship between the spectral bounds and the Turán property, we will first consider eigenalues of the (normalized) Laplacian. Detailed definitions will be gien in the next section. The connection between eigenalues of the Laplacian and Turán numbers depends on a notion of generalized olumes: For a subset X of ertices in a graph G, thek-olume of X is defined by ol k (X) = d k, where denotes the degree of in G. We will first describe seeral key properties of graphs which are consequences of spectral gaps. In particular, we will gie seeral general isoperimetric inequalities in Section 3. These inequalities proide good estimates for the discrepancies of a graph. We will use these inequalities to establish the relationship between eigenalues and the Turán property. We will show that if the non-zero eigenalues of the (normalized) Laplacian are bounded (depending on t and the olumes of G), then the graph is t-turán. The proofs are gien in Section 4. 2 Preliminaries on eigenalues For a graph G, there are seeral ways to ealuate eigenalues by associating arious matrices with G. A typical matrix is the adjacency matrix A = A G which has entries A(u, ) =ifuan are adjacent, and 0 otherwise. Another matrix is the combinatorial Laplacian L which is defined as L = D A where D is the diagonal matrix with diagonal entries D(, ) = where is the degree of the ertex. The well-known matrix-tree theorem of Kirchhoff [4] states that the number of spanning trees in a graph G is the product of all (except for the smallest) eigenalues of L diided by the number of ertices of G. The eigenalues of the adjacency matrix are useful in enumerating walks in a graph. For example, the largest eigenalue of A, denoted by A, is the limit of the k-th root of the number of k-walks in G, askapproaches infinity. In this paper, we will mainly focus on the (normalized) Laplacian L, which is defined as follows: A(,) if u = and 0 L(u, ) = if u an are adjacent, du 0 otherwise. 2

3 We can write L = D /2 LD /2 with the conention D (, ) =0for =0. For a regular graph with degree d, wehae L=I d A. Let g denote an arbitrary function which assigns to each ertex of G a real alue g(). We can iew g as a column ector. Then g, Lg g, g = g, D /2 LD /2 g g, g f,lf = D /2 f,d /2 f u (f(u) f())2 = (2) f()2 where g = D /2 f and u denotes the sum oer all unordered pairs {u, } for which u an are adjacent. Here f,g = x f(x)g(x) denotes the standard inner product in Rn. (We note that we can also use the inner product f,g = f(x)g(x) for complex-alued functions.) From equation (2), we see that all eigenalues are non-negatie and 0 is an eigenalue of L. We denote the eigenalues of L by 0 = λ 0 λ λ n. Let denote the constant function which assumes the alue on each ertex. Then D /2 is an eigenfunction of L with eigenalue 0. Quite a few basic facts can be deried from the aboe definition (see [2]). All λ i are between 0 and 2. The number of eigenalues of L haing alue 0 is the same as the number of connected components in G. The maximum eigenalue of L is 2 if and only if the graph is bipartite. In the next few sections, we will focus on the family F δ of graphs with Laplacian eigenalues satisfying λ =max i 0 λ i <δ (3) for i 0. We note that for d-regular graphs, the eigenalues of the adjacency matrix are just d( λ i )so the so-called (n, d, λ)-graphs in [, 5] are in F δ for δ = λ/d. 3 Eigenalues and discrepancies A main tool for inestigating arious graph inariants for F δ concerns the notion of discrepancy and the related discrepancy inequalities. A typical definition for discrepancy is the difference between the actual 3

4 quantity and the expectealue. The goal is to upper bound the discrepancy in terms of eigenalues. For example, in a gien graph G, a quantity of concern is the number e(x, Y ) of edges between two subsets X and Y. In many situations (such as G is regular), the expectealue of e(x, Y ) is taken to be the edge density multiplied by the cardinality of X and Y. The condition of the graph being regular is quite restrictie. In particular, such an inequality cannot be applied to (non-regular) subgraphs of a regular graph. Here we extend such a discrepancy inequality to general graphs by using the eigenalues of the Laplacian. Lemma Suppose a graph G on n ertices has eigenalues λ i of the Laplacian satisfying λ =max i 0 λ i <δ. For any two subsets X and Y of ertices, e(x, Y ) denotes the number of ordered pairs (x, y) so that {x, y} is an edge and x X and y Y. Then e(x, Y ) satisfies where ol(x) = x X d x and =. e(x, Y ) ol(x)ol(y ) δ ol(x)ol(y ) The aboe lemma is a special case of the following: Lemma 2 Suppose k is a gien real alue (possibly negatie) and a graph G on n ertices has Laplacian eigenalues λ i satisfying λ =max i 0 λ i <δ. Then for any two subsets X and Y of ertices, the k-weight of X and Y, denoted by e k (X, Y )= d k u dk u X Y, u satisfies e k (X, Y ) ol k+(x)ol k+ (Y ) δ ol 2k+ (X)ol 2k+ (Y ) where ol i (X) = x X di x anol i(g) = di. Proof: Then We define ψ X (u) = { d k u if u X, 0 otherwise. e k (X, Y ) ol k+(x)ol k+ (Y ) = ψ X,Aψ Y ol k+(x)ol k+ (Y ) = ψ X,D /2 (I L)D /2 ψ Y ol k+(x)ol k+ (Y ) = ψ X,D /2 (I L φ 0φ 0 )D /2 ψ Y 4

5 since the eigenector φ 0 associated with eigenalue 0 has coordinates /, where f denotes the transpose of f. Since G is in F δ,wehae I L φ 0φ 0 δ. Therefore e k (X, Y ) ol k+(x)ol k+ (Y ) D /2 ψ X I L φ 0φ 0 D /2 ψ Y = ol 2k+ (X) δ ol 2k+ (Y ) as desired. Foraertexin a graph G, the neighborhood Γ() ofis defined by Γ() ={u : u }={u : {u, } is an edge.} In general, the neighborhood Γ X () ofin X is denoted by Γ X () ={u X : u }. In addition to Lemma 2, we also need the following estimate: Lemma 3 Suppose a graph G on n ertices has Laplacian eigenalues λ i satisfying λ =max i 0 λ i <δ. For any real alue k and any subset X of ertices of G, we hae where Γ X () ={u X:u }. ( ) 2 ol k+ (X) ol k (Γ X ()) δ 2 ol 2k+(X) Proof: We consider ψ X as defined in the proof of Lemma 2. The difference of (I L)D /2 ψ X and the projection of D /2 ψ X on D /2 can be written as: D /2 Aψ X D /2 ψ X,D /2 D/2 δ D/2 ψ X which implies ( ) 2 ol k+ (X) ol k (Γ X ()) δ 2 ol 2k+(X) as desired. 5

6 Lemma 4 Suppose a graph G on n ertices has Laplacian eigenalues λ i satisfying λ =max i 0 λ i <δ. Suppose X is a subset of ertices of G an is a ertex in X. LetΓ X ()denote the neighborhood of in X and let R() denote a subset of Γ(). We hae and Γ X () 2 ol3 (X) ol 2 (G) + O(δ ol2 (X) )+O(δ2 ol(x)). (4) Γ X () R() R() ol(x) + O(δ ol2 (X) )+O(δ2 ol(x)) (5) Proof: Using Lemma 3, we hae Γ X () 2 = ol 2 (X) ol 2 + (G) Γ X () 2 d 2 ol2 (X)/ol 2 (G) ol3 (X) ol 2 (G) + O(δ ol2 (X) )+O(δ2 ol(x)). Γ X () R() = R() ol(x) + R() ( Γ X () ol(x)/) R() ol(x) R() ol(x) + δ ol(x)( R() 2 ) + O(δ ol(x)2 )+O(δ2 ol(x)). A useful generalization of Lemma 4 is the following: Lemma 5 Suppose a graph G on n ertices has Laplacian eigenalues λ i satisfying λ =max i 0 λ i <δ. Suppose X is a subset of ertices of G and i is a non-negatie alue. We hae and d i+ (ol i (Γ X ())) 2 = ol3 i+(x) ol 2 (G) + O(δ ol i+(x)ol 2i+ (X) ) d i+ ol i (Γ X ())ol i (R()) = ol i+(x) d i {u,}red di u + O( λ ol i+(x)ol 2i+ (X) ). 6

7 Proof: d i+ d i+ (ol i (Γ X ())) 2 = ol3 i+(x) ol 2 (G) = ol3 i+ (X) ol 2 (G) ol i (R())ol i (Γ X ()) = d i + + ol 2 i(γ X ()) ( ol i+ (X)/) 2 d i+ + O( λ ol i+(x)ol 2i+ (X) ), ol i (R()) ol i+(x) d i+ = ol i+(x) ol i (R())(ol i (Γ X ()) ol i+ (X) ) d i {u,}red di u + O( λ ol i+(x)ol 2i+ (X) ). Lemma 6 Suppose that X is a subset of ertices in a graph G and α β are non-negatie alues. Then ol α (X)ol β (X) ol α+ (X)ol β (X). (6) Proof: The inequality (6) follows from the following general ersion of the Cauchy-Schwarz inequality for positie a j s and 0 α β : k k k ( a α j )( a β j ) ( a α j )( j= j= j= k j= a β+ j ). By choosing a j s to be the reciprocal of the degrees, (6) is an immediate consequence. 4 A generalization of Turán s theorem We will now proe the main theorem: Theorem Suppose a graph G on n ertices has eigenalues 0 = λ 0 λ... λ n with λ = max i 0 λ i satisfying λ = o( ). ol 2t+3 (G) t 2 (7) Then, G is t-turán for t 2; i.e., any subgraph of G containing no K t+ has at most ( /t + o())e(g) edges where e(g) is the number of edges in G. 7

8 There are expressions in terms of o( ) s in the statement of Theorem. To be precise, the result in Theorem can be restated as follows: For any ɛ>0,thereisaδsuch that if the eigenalues of the Laplacian of the graph G satisfies λ =max i 0 λ i < δ ol 2t+3 (G) t 2, then any subgraph of G containing no K t+ has at most ( /t + ɛ)e(g) edges where e(g) is the number of edges in G. We remark that the condition in (7) has the following implication for the minimum degree η of G. λ < ol 2t+3 (G) t 2 η 2t 3 t 2 n η2t 3 ηt n t = ηt 2 n t Since the inequality always holds, condition (7) implies that λ / ηn η n (2t 3)/(2t ). Thus, condition (7) may hold only if the minimum degree of the graph G is sufficiently large. Theorem implies the following two facts the Classical Turán therem and the case for regular graphs, Corollary A graph on n ertices containing no K t+ has at most ( /t + o())n 2 /2 edges. Proof: The complete graph on n ertices has Laplacian eigenalues 0 and n/(n ) (with multiplicity n ). Thus, for G = K n, it is always true that λ = n = o(). Therefore, Theorem implies that any graph on n ertices containing no K t+ has at most ( /t+o())n 2 /2 edges. Corollary 2 If a graph is regular with degree d and has n ertices, then the condition in (7) is just λ = o (( dn ) )t. Then any subgraph of G containing no K t+ has at most ( /t + o())dn/2 edges. 8

9 The proof is by directly substitution and will be omitted. Suppose that R is a subset of edges so that eery K t+ in G contains at least one edge in R. In order to proe Theorem, we wish to show that R ( + o()) E(G) /t. To do so, we will proe the following stronger result: Theorem 2 Suppose a graph G on n ertices has eigenalues satisfying (7). Then we hae (*) For any subset X of ertices in G and for non-negatie integer k t, ifrcontains an edge from eery complete subgraph on k + ertices, then we hae, for all i, 0 i k, ol2 k i+(x) + O λ ol 2i 2j+ (X)ol j (G) (8) k d i,u X,{u,} R u di To derie Theorem from Theorem 2, suppose that R contains an edge from eery complete subgraph on k + ertices. We apply (*) with i =0andk=t. We then use (7) and hae R t t + O( λ ol 2j+ (G)ol j (G)) ( + O( λol 2t+3 (G)ol t 2 (G))) t ( + o()). t Therefore G is t-turán, as claimed. 5 Proofs of the main theorems In this section, we shall proe Theorem 2. The inequality in (*) is somewhat complicated. We shall deal with simpler cases first (such as i = 0) which contain the main ideas. Proof of Theorem 2: First we want to show that (*) holds for k =. In this case we hae R = E(G). Lemma 2 implies that (*) holds for k =. Suppose that k 2 and (*) holds for k <k. We wish to proe (*) for k. Suppose R contains an edge from eery complete subgraph on k + ertices. We want to show that the inequality (8) holds for all i, 0 i k. We shall first proe the case that i = 0. Suppose that edges in R are colored red and the rest of the edges of G are blue. We focus on edges inside of the gien set X. Recall that Γ X () =Γ() X. For each ertex, letr() andb() denote the set of neighbors u of in X with {u, } red and blue, respectiely. For a 9

10 ertex, we consider the induced subgraph on B() which does not contain a complete graph on k ertices. By induction we hae B() 2 (k ) + O λ ol 2j (B())ol j (G). (9) u,w B(),{u,w}red We consider the set T j of all triangles in X containing exactly j rededges,forj=,2,3. We note that Thus, 2 W = {u,,w} T = By Lemma 2, we hae Therefore, u,w B(),{u,w}red u R() w Γ X() Γ(u) u R() w Γ X() Γ(u) W u R() w Γ X(),w u u,w R(),u w W W 2 +3W 3 = u R() w Γ X(),w u u R() w Γ X(),w u 2 {,u,w} T 2 2W 2 3W 3 u,w R(),u w. W 2 3W 3 u,w R(),u w 3 {,u,w} T 3. = R() Γ X() + O( λ ol R()ol (Γ X ())) = R() 2 + O( λol (R()). Γ X () R() R() 2 + O( λ ol (Γ X ()) ). (0) 0

11 On the other hand, from (0) we hae W = 2 {u,,w} T B() 2 (k ) O( λ ol 2j (B())ol j (G) ) d B() 2 (k ) O( λ ol 2j (Γ X ())ol j (G) ) d (k ) ( Γ X () R() ) 2 O( λ +O( λ 2 ol (X)ol 4j (X)ol j (G)). X ol 2j (X)ol j (G) ) We note that the terms inoling λ 2 are of lower order by using the assumption on λ. Combining the preceding upper and lower bounds for W,wehae (k+) Γ X () R() Γ X () 2 +k R() 2 O λ X ol 2j (X)ol j (G). () By Lemma 4 and inequality (5), we hae Γ X () R() R() ol(x) + O( λ ol(x)2 )+O( λ 2 ol(x)) Also, by Lemma 4 and inequality (4), we hae Γ X () 2 ol3 (X) ol(x)2 ol 2 + O( λ (G) )+O( λ 2 ol(x)). Substituting into (), we hae (k +) R() ol(x) ol3 (X) ol 2 (G) + k R() 2 ol(x) + O λ( 2 d + X ol 2j (X)ol j (G)) ol3 (X) ol 2 (G) + k ( R() )2 ol(x) d + O λ( 2 + X ol 2j (X)ol j (G)). This implies ( ol2 (X) R() )( ol2 (X) k ol(x) R() )+O λ( 3 + X ol 2j (X)ol j (G)ol(X) < 0.

12 Thus we hae R = R() 2 ol 2 (X) 2k + O X ol 2j (X)ol λ(ol(x)+ j+ (G) ) ol(x) ol 2 (X) 2k + O λ(ol(x)+ ol 2j (X)ol j+ (G)) k k E(X) + O λ(ol(x)+ ol 2j+ (X)ol j (G)) j= k k E(X) + O λ ol 2j+ (X)ol j (G). We hae completed the proof for the case i =0. Suppose i. For j =,2,3, we consider W (i) j = 2 d i. {u,,w} T di u di w j As before, we hae W (i+) d i+ d i+ u R() w Γ X() Γ(u) u R() w Γ X() Γ(u) d i+ u d i+ W (i+) 2 3W (i+) 3 w d i+ u d i+ w ol i (R())ol i (Γ X ()) d i+ d i+ u,w R(),u w d i+ u d i+ w ol 2 i(r()) d i+ + O( λ ol i (R())ol i (Γ X ()) d i+ ol 2 i (R()) d i+ +O( λ ol i(x)ol 2i (X) + λ 2 (ol 2i (X)ol 4i (X)) /2 ). ol 2i (Γ X ()) d i+ ) 2

13 On the other hand, by induction we hae W (i+) = 2 {u,,w} T d (i+) d (i+) u d (i+) w (k ) (k ) +O λ d (i+) ol2 i(b()) + O( λ ol 2 i(γ X ()) ol i (R()) d i+ ol i (X)ol 2i 2j (X)ol j (G) ol 2i 2j+ (B())ol j (G) + λ 2 ol 2i (X)ol 4i 4j (X)ol j (G). By combining the upper and lower bounds of W (i+) (and multiplying by ), we hae where A, B, C, D are as follows: A = d i+ (ol i (Γ X ())) 2 (k +)A+BkC + O( λd) (2) ol3 i+ (X) ol 2 + O( λ ol i+(x)ol 2i+ (Γ()) ) by using Lemma 5. (G) B = d i+ ol 2 i (R()) ( ol i (R()) d i ) 2 /ol i+ (G) by the Cauchy-Schwarz inequality. C = D = d i+ ol i+(x) ol i (Γ X ())ol i (R()) d i {u,}red di u + O( λ ol i+(x)ol 2i+ (X) ) ol i (X)ol 2i 2j (X)ol j (G)+ol i+ (X)ol 2i+ (X) By substituting A, B and C into (2), we hae the following: This implies k ol3 i+ (X) ol 2 (G) d i {u,}red di u + ( {u,}red d ) 2 i di u (k +) ol i+(x) ol i+ (X) ol2 i+ (X) k O λ(ol 2i+ (X)+ d i {u,}red di u + O ( λd ). ol i (X)ol 2i 2j (X)ol j+ (G) ol i+ (X) ). 3

14 Now, by using Lemma 6 and inequality (6), we hae d i d i u {u,}red ol2 i+(x) k ol2 i+(x) k + O( λol 2i+ (X)+ ol 2i 2j (X)ol j+ (G)) k + O( λ ol 2i 2j+ (X)ol j (G)). This completes the proof of Theorem 2. References [] N. Alon, Eigenalues and expanders, Combinatorica 6 (986), [2] F. Chung, Spectral Graph Theory, AMS Publications, 997. [3] F. Chung, L. Lu and V. Vu, Spectra of random graphs with gien expected degrees, Proceedings of National Academy of Sciences, 00, no., (2003), [4] F. Kirchhoff, Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galanischer Ströme geführt wird, Ann. Phys. chem. 72 (847) [5] M. Krieleich and B. Sudako, Pseudo-random graphs, preprint. [6] B. Sudako, T. Szabo and V. Vu, A generalization of Turán s theorem, J. Graph Theory, 49 (2005),