Upper bound on the $k$-th eigenvalue of a graph
We prove a general upper bound on the $k$-th adjacency eigenvalue of a graph. For $k\ge 2$, we show that \[ λ_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1 \] for every graph $G$ on $n$ vertices. We build on a recent approach that addresses the case…
Authors: Varun Sivashankar
UPPER BOUND ON THE k -TH EIGENV ALUE OF A GRAPH V AR UN SIV ASHANKAR Abstra ct. W e pro v e a general upp er bound on the k -th adjacency eigen v alue of a graph. F or k ≥ 2 , w e sho w that λ k ( G ) ≤ ( k − 2) √ k + 1 + 2 2 k ( k − 1) n − 1 for ev ery graph G on n v ertices. W e build on a recent approac h that addresses the case k = 3 and generalize the upp er b ound for all k ≥ 3 b y using the p ositivit y of Gegen bauer p olynomials. The upp er bound is tigh t for k ∈ { 2 , 3 , 4 , 8 , 24 } . W e also highligh t the close relation of λ k ( G ) to questions ab out equiangular lines. 1. Intr oduction F or a simple undirected graph G on n vertices, let λ 1 ( G ) ≥ λ 2 ( G ) ≥ · · · ≥ λ n ( G ) denote the eigen v alues of its adjacency matrix. F ollowing Nikiforo v [ Nik15 ], define c k := sup λ k ( G ) | V ( G ) | : | V ( G ) | ≥ k . The problem of b ounding λ k ( G ) in terms of the order of the graph go es back at least to w ork of Hong and Po wers [ Hon88 , Hon93 , P o w89 ]. Nikiforo v [ Nik15 ] prov ed the univ ersal estimate c k ≤ 1 2 √ k − 1 ( k ≥ 2) , On the lo w er-b ound side, the balanced complete k -partite graph giv es c k ≥ 1 /k , and Nikiforo v show ed that in fact c k > 1 /k for every k ≥ 5 [ Nik15 ]. He therefore ask ed whether c 3 = 1 / 3 and c 4 = 1 / 4 [ Nik15 , Question 2.11]. P o wers had earlier prop osed a conjecture that λ k ( G ) ≤ ⌊ n k ⌋ for connected graphs, but Nikiforo v later observ ed that the original pro of is flaw ed; moreov er, Nikiforov’s constructions show that this b ound fails for ev ery k ≥ 5 , while Linz later ruled out the case k = 4 [ P o w89 , Nik15 , Lin23 ]. The remaining case k = 3 is also recorded in the survey of Liu and Ning [ LN23 , Problem 10]. Recen t work has clarified the first unresolved cases. Leonida and Li verified the sharp third-eigen v alue bound for sev eral imp ortan t graph classes and form ulated a w eighted matrix conjecture that would imply it in full generality [ LL25 ]. See also Li [ Li25 ] for a related asymptotic improv emen t on Nikiforov’s bound. V ery recently , T ang prov ed the sharp inequalit y λ 3 ( G ) ≤ n 3 − 1 for every graph G b y confirming the conjecture of Leonida and Li [ T an26 ]. On the low er-b ound side, it is easy to see that c 3 ≥ 1 / 3 by taking the disjoin t union of three equal sized cliques. [ Nik15 ] prov ed that c k > 1 2 √ k − 1+1 when q is an o dd prime Princeton University. Email: varunsiva@princeton.edu . 1 2 V ARUN SIV ASHANKAR p o wer and k = q 2 − q + 1 . [ Nik15 ] also sho wed that c k > 1 2 √ k − 1+ k 1 / 3 for sufficiently large k . F urther, Linz pro vided several impro ved constructions for 4 ≤ k ≤ 24 [ Lin23 ]. Our main result is the following. Theorem 1.1. L et k ≥ 2 b e an inte ger. F or every gr aph G on n vertic es, λ k ( G ) ≤ α k n − 1 , α k := ( k − 2) √ k + 1 + 2 2 k ( k − 1) Conse quently, c k ≤ α k F or k = 2 , this simply follows from the classical sharp b ound λ 2 ( G ) ≤ n/ 2 − 1 [ Hon88 , Nik15 ]. Our result holds for k ≥ 3 . F or k = 3 , it reco v ers T ang’s sharp b ound λ 3 ( G ) ≤ n/ 3 − 1 [ T an26 ]. F or k ∈ { 4 , 8 , 24 } , α k matc hes Linz’s lo w er b ound [ Lin23 ]. Therefore, our upp er b ound is tight for k ∈ { 2 , 3 , 4 , 8 , 24 } . It is v ery interesting to note that our upp er b ound for λ k ( G ) is tight whenev er an extremal equiangular lines configuration exists in dimension r = k − 1 . W e discuss this in further detail in Section 5.1 . F or every k ≥ 2 one has α k < 1 2 √ k − 1 , so Theorem 1.1 improv es Nikiforo v’s universal b ound for λ k ( G ) . W e remark that Nikiforov claimed that there exists a p ositive ϵ k suc h that c k ≤ 1 2 √ k − 1 − ϵ k . How ev er, the pro of was omitted b ecause it relied on the Remov al Lemma of Alon, Fisc her, Krivelevic h, and Szegedy [ AFKS00 ] with other to ols of analytic graph theory . This w ould make the pro of complicated and give a small ϵ k . Our pro of follo ws the ov erall strategy of T ang [ T an26 ]. As in their work, the graph- theoretic theorem is deduced from a w eigh ted matrix inequality using the iden tit y A ( G ) + A ( G ) = J − I and W eyl’s i nequalit y . The k ey new mo dification is the use of Gegen bauer p olynomi- als to generalize the trigonometric upp er b ound from the rank- 2 case. The required p ositiv e-kernel input go es bac k to Sc ho enberg [ Sc h42 ]; for related p ersp ectiv es in spher- ical co ding and semidefinite metho ds, see also Delsarte–Go ethals–Seidel [ DGS77 ] and Bac ho c–V allen tin [ BV08 ]. 2. Positive Kernels Fix an in teger r ≥ 2 . Let S r − 1 = { v ∈ R r : ∥ v ∥ 2 = 1 } b e the unit sphere. F or real v ectors u, v , let ⟨ u, v ⟩ denote the standard inner pro duct. F or real v alued square matrices A, B ∈ R n × n , write ⟨ A, B ⟩ F = tr ( A T B ) for the F rob enius inner pro duct. This is equiv alent to just taking taking the inner pro duct b y viewing A and B as n 2 -dimensional vectors. F or a matrix, ∥ A ∥ F = p ⟨ A, A ⟩ F and ∥ A ∥ 1 = P i,j | A ij | . Let I r ∈ R r × r denote the iden tit y matrix. A matrix M ∈ R n × n is p ositiv e semidefinite if x ⊤ M x ≥ 0 for all x ∈ R n . Definition 2.1. A function f : R → R is p ositiv e semidefinite on S r − 1 if for all unit v ectors u 1 , . . . , u n ∈ S r − 1 , the matrix M ∈ R n × n giv en by M ij = f ( ⟨ u i , u j ⟩ ) is p ositiv e semidefinite. Define f r 2 ( t ) = t 2 − 1 r f r 4 ( t ) = t 4 − 3 r ( r + 2) UPPER BOUND ON THE k -TH EIGENV ALUE OF A GRAPH 3 W e will pro v e that f r 2 and f r 4 as p ositiv e semidefinite on S r − 1 . In Section 3 , this will pla y a crucial role in b ounding ∥ Q ∥ 1 for a rank r orthogonal pro jection matrix Q ∈ R n × n . Lemma 2.2. F or r ≥ 2 , define ϕ ( u ) := uu ⊤ − 1 r I r , u ∈ S r − 1 . Then for al l u, v ∈ S r − 1 , ⟨ ϕ ( u ) , ϕ ( v ) ⟩ F = f r 2 ( ⟨ u, v ⟩ ) . F urther, f r 2 is p ositive semidefinite on S r − 1 . Pr o of. A direct computation gives ⟨ ϕ ( u ) , ϕ ( v ) ⟩ F = tr uu ⊤ − 1 r I r v v ⊤ − 1 r I r = ( u ⊤ v ) 2 − 1 r = ⟨ u, v ⟩ 2 − 1 r = f r 2 ( ⟨ u, v ⟩ ) . No w observ e that for any u 1 , . . . , u n ∈ S r − 1 , the n × n matrix ( f r 2 ( ⟨ u i , u j ⟩ )) n i,j =1 is just the Gram matrix of the matrices ϕ ( u 1 ) , . . . , ϕ ( u n ) with the standard F rob enius inner pro duct. Therefore, it m ust b e p ositive semidefinite. □ T o prov e that ( f r 2 ( ⟨ u i , u j ⟩ )) n i,j =1 is PSD, we expressed it as a Gram matrix of certain matrices with the F rob enius inner pro duct. W e will do the same thing for f r 4 , but since f r 4 is a degree 4 p olynomial, we will need to work with 4 -tensors. The pro of of Lemma 2.3 is elementary but requires some computation. W e defer this pro of to the app endix. Lemma 2.3. F or r ≥ 2 , f r 4 ( t ) = t 4 − 3 r ( r + 2) is p ositive semidefinite on S r − 1 . Remark 2.4. Lemma 2.2 and Lemma 2.3 are just sp ecial cases of a theorem by Sc ho en b erg[ Sc h42 ]. Let G λ ℓ denote the Gegen bauer polynomial of degree ℓ . These p olynomials can b e defined in terms of their generating function[ SW71 ]: 1 (1 − 2 tx + x 2 ) λ = ∞ X ℓ =0 G λ ℓ ( t ) x ℓ Sc ho enberg pro ved the following strong characterization: f ( t ) is a real con tinuous function suc h that the matrix f ( ⟨ u i , u j ⟩ ) n i,j =1 is p ositive semidefinite for all subsets { u 1 , . . . , u n } ⊆ S r − 1 if and only if f is of the form P ∞ ℓ =0 a ℓ G r/ 2 − 1 ℓ with a ℓ ≥ 0 . In particular, G r/ 2 − 1 2 and G r/ 2 − 1 4 are p ositiv e semidefinite on S r − 1 , and w e are simply using f r 2 = r − 1 r G r/ 2 − 1 2 and f r 4 = ( r − 1)( r +1) ( r +2)( r +4) G r/ 2 − 1 4 + 6 r +4 f r 2 . Since f 4 r and f 4 r are just positive linear combinations of Gegenbauer p olynomials with λ = r / 2 − 1 , they must b e p ositiv e semidefinite on S r − 1 b y [ Sc h42 ]. How ever, w e do not require this strong c haracterization. 4 V ARUN SIV ASHANKAR 3. An ℓ 1 bound f or rank- r or thogonal pr ojections Let Q = ( q ij ) ∈ R n × n b e a rank- r orthogonal pro jection, where r ≥ 2 . So Q satisfies Q = Q 2 = Q T . The goal of this section is to determine an upp er b ound on ∥ Q ∥ 1 = P i,j | q ij | . It is easy to see that for a rank r orthogonal pro jection Q , ∥ Q ∥ 1 ≤ √ n 2 ∥ Q ∥ F = n √ r . In [ MMP19 ], they prov e that ∥ Q ∥ 1 ≤ r + p ( n − 1) r ( n − r ) . Our b ound in Theorem 3.1 is b etter precisely when n ≥ r +1 2 with equality at n = r +1 2 . W e require this impro ved b ound b ecause r will b e small relative to n in our applications. W e prov e the following: Theorem 3.1. L et Q ∈ R n × n b e a r ank- r ortho gonal pr oje ction, wher e r ≥ 2 . ∥ Q ∥ 1 = n X i,j =1 | q ij | ≤ β r n wher e β r := r + √ r + 2 1 + √ r + 2 Pr o of. Since Q is a rank r orthogonal pro jection, we ma y write Q = B B ⊤ suc h that B ∈ R n × r where the columns of B form an orthogonal set in R n . F urther, let x 1 , . . . , x n ∈ R r b e the ro ws of B . W rite x i = c i u i , where c i = ∥ x i ∥ 2 ≥ 0 and u i ∈ S r − 1 whenev er c i = 0 ; if c i = 0 , choose u i arbitrarily in S r − 1 . Since the columns of B are orthogonal, B ⊤ B = I r . W riting out B ⊤ B in terms of the ro ws of B , w e obtain (1) n X i =1 c 2 i u i u ⊤ i = I r and q ij = c i c j ⟨ u i , u j ⟩ Set C := n X i =1 c i , x := n X i =1 c i ϕ ( u i ) , X := ∥ x ∥ F . T o pro ve Theorem 3.1 , w e establish Lemma 3.2 , Lemma 3.3 and Lemma 3.4 . Lemma 3.2. With the notation ab ove, C 2 + r X 2 ≤ r n. Pr o of. C 2 + r X 2 = n X i =1 n X j =1 c i c j + r n X i =1 n X j =1 c i c j ⟨ ϕ ( u i ) , ϕ ( u j ) ⟩ F = n X i =1 n X j =1 c i c j + r n X i =1 n X j =1 c i c j ⟨ u i , u j ⟩ 2 − 1 r b y Lemma 2.2 = r n X i =1 n X j =1 c i c j ⟨ u i , u j ⟩ 2 ≤ r 2 n X i =1 n X j =1 ( c 2 i + c 2 j ) ⟨ u i , u j ⟩ 2 b y Cauch y-Sc h warz = r n X i =1 n X j =1 c 2 j ⟨ u i , u j ⟩ 2 b y symmetry UPPER BOUND ON THE k -TH EIGENV ALUE OF A GRAPH 5 r n X i =1 n X j =1 c 2 j ⟨ u i , u j ⟩ 2 = r n X i =1 n X j =1 tr( u i u ⊤ i c 2 j u j u ⊤ j ) = r n X i =1 tr u i u ⊤ i n X j =1 c 2 j u j u ⊤ j ! = r n X i =1 tr u i u ⊤ i b y Equation (1) = r n □ W e will no w b ound ∥ Q ∥ 1 b y finding a p olynomial h ( t ) that satisfies | t | ≤ h ( t ) for all t ∈ [ − 1 , 1] and applying it entry wise to | q ij | = c i c j |⟨ u i , u j ⟩| with t = |⟨ u i , u j ⟩| . W e will consider an expression of the form a + b f r 2 ( t ) − γ f r 4 ( t ) . This is b ecause after substituting t = ⟨ u i , u j ⟩ and summing ov er i, j , the constant term pro duces C 2 , the f r 2 -term pro duces X 2 , while the f r 4 -term is nonnegative since f r 4 is p ositiv e semidefinite by Lemma 2.3 . The following lemma formalizes this reduction. Lemma 3.3. L et a, b, γ ∈ R satisfy | t | ≤ a + b f r 2 ( t ) − γ f r 4 ( t ) for al l t ∈ [ − 1 , 1] , with γ ≥ 0 and b ≤ r a . Then every r ank- r ortho gonal pr oje ction Q ∈ R n × n satisfies ∥ Q ∥ 1 ≤ ar n. Pr o of. Recall b y ( 1 ) that w e may write Q = B B T where the ro ws of B are giv en by c i u i for i ∈ [ n ] and ∥ Q ∥ 1 = P i,j | q ij | = P i,j c i c j |⟨ u i , u j ⟩| . Applying the assumed scalar inequalit y with t = ⟨ u i , u j ⟩ , using the fact that c i ≥ 0 and summing o ver all i, j , w e obtain ∥ Q ∥ 1 = X i,j | q j | ≤ a X i,j c i c j + b X i,j c i c j f r 2 ( ⟨ u i , u j ⟩ ) − γ X i,j c i c j f r 4 ( ⟨ u i , u j ⟩ ) . By definition, X i,j c i c j = C 2 , X i,j c i c j f r 2 ( ⟨ u i , u j ⟩ ) = X i,j c i c j ⟨ ϕ ( u i ) , ϕ ( u j ) ⟩ F = X 2 Also, the p ositiv e semidefiniteness of f r 4 ( ⟨ u i , u j ⟩ ) n i,j =1 implies that X i,j c i c j f r 4 ( ⟨ u i , u j ⟩ ) ≥ 0 . Therefore ∥ Q ∥ 1 ≤ aC 2 + bX 2 . Since b ≤ r a , we ha ve aC 2 + bX 2 ≤ a C 2 + r X 2 . No w apply Lemma 3.2 to get ∥ Q ∥ 1 ≤ ar n. □ 6 V ARUN SIV ASHANKAR W e now exhibit a concrete choice of co efficients for which the h yp otheses of Lemma 3.3 hold. Clearly , we would lik e ar to b e as small as p ossible. The c hoice of co efficien ts in Lemma 3.4 b elo w ma y seem m ysterious but it is motiv ated b y a connection to equiangular lines. A more detailed discussion is included in Section 5 . Lemma 3.4. L et r ≥ 2 , and set s := √ r + 2 . Define a r := s 2 + s − 2 r ( s + 1) , b r := s ( s 2 + 2 s + 3) 2( s + 1) 2 , γ r := s 3 2( s + 1) 2 . Then for every t ∈ [ − 1 , 1] , | t | ≤ a r + b r f r 2 ( t ) − γ r f r 4 ( t ) . Mor e over, b r ≤ r a r . Pr o of. Recall that f r 2 ( t ) = t 2 − 1 r , f r 4 ( t ) = t 4 − 3 r ( r + 2) . Since the left-hand side is even in t , it suffices to consider t ∈ [0 , 1] . A direct expansion giv es a r + b r f r 2 ( t ) − γ r f r 4 ( t ) − t = (1 − t )( st − 1) 2 ( st + s + 2) 2( s + 1) 2 . The right-hand side is nonnegative for t ∈ [0 , 1] , since 1 − t ≥ 0 , ( st − 1) 2 ≥ 0 , st + s + 2 > 0 . Hence | t | ≤ a r + b r f r 2 ( t ) − γ r f r 4 ( t ) for all t ∈ [ − 1 , 1] . Finally , r a r − b r = s 3 + 2 s 2 − 5 s − 4 2( s + 1) 2 = ( s − 2)( s + 1)( s + 3) + 2 2( s + 1) 2 ≥ 0 , and therefore b r ≤ r a r . □ Completing the pr o of of The or em 3.1 . Apply Lemma 3.3 with a = a r , b = b r , γ = γ r . By Lemma 3.4 , the h yp otheses of Lemma 3.3 are satisfied. Therefore ∥ Q ∥ 1 ≤ a r r n = s 2 + s − 2 s + 1 n = r + √ r + 2 1 + √ r + 2 n = β r n. □ UPPER BOUND ON THE k -TH EIGENV ALUE OF A GRAPH 7 4. Fr om projections to eigenv alues Theorem 4.1. L et A = ( a ij ) ∈ R n × n b e symmetric, with eigenvalues µ 1 ≥ µ 2 ≥ · · · ≥ µ n . Assume that 0 ≤ a ij ≤ 1 ( i = j ) , a ii ≥ 0 (1 ≤ i ≤ n ) . L et r ≥ 2 b e an inte ger. Then µ n − r +1 + · · · + µ n ≥ − β r 2 n. In p articular, µ n − r +1 ≥ − β r 2 r n. Pr o of. Let P r denote the set of rank- r orthogonal pro jections in R n × n . By the v ariation c haracterization of eigenv alues (Ky F an’s principle), it is easy to see that λ 1 ( M )+ · · · + λ r ( M ) = max B ∈ R n × r : B T B = I r tr( B T M B ) = max B ∈ R n × r : B T B = I r tr( M B B T ) = max Q ∈P r tr( M Q ) W e simply apply this to M = − A to obtain: µ n − r +1 + · · · + µ n = min Q ∈P r tr( AQ ) . Fix Q = ( q ij ) ∈ P r . Since Q is p ositiv e semidefinite, w e ha v e 1 ⊤ Q 1 ≥ 0 . F urther, tr ( Q ) = tr ( B B T ) = tr ( B T B ) = r . This is also evident b ecause Q is a rank r orthogonal pro jection and so has r eigenv ectors with eigenv alue 1 . X i
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