Non-bipartite distance-regular graphs with a small smallest eigenvalue
aa r X i v : . [ m a t h . C O ] J a n Non-bipartite distance-regular graphs with a smallsmallest eigenvalue
Zhi Qiao a , Yifan Jing b , and Jack Koolen ∗ ca School of Mathematical Sciences, Sichuan Normal University, 610068,Sichuan, PR China b Department of Mathematics, University of Illinois at Urbana Champaign,Urbana, IL, 61801, USA c School of Mathematical Sciences, University of Science and Technology ofChina, Wen-Tsun Wu Key Laboratory of the Chinese Academy ofSciences, 230026, Anhui, PR China
Abstract
In 2017, Qiao and Koolen showed that for any fixed integer D ≥
3, there are onlyfinitely many such graphs with θ min ≤ − αk , where 0 < α < θ min compared with k . In particular, we will show that if θ min is relatively closeto − k , then the odd girth g must be large. Also we will classify the non-bipartitedistance-regular graphs with θ min ≤ D − D for D = 4 , Key words : Distance-regular graphs, Smallest eigenvalue, Odd girth
AMS classification : 05C75, 05E30, 05C50
The odd girth of a non-bipartite graph is the length of its shortest odd cycle. Let Γ be anon-bipartite distance-regular graph with valency k , diameter D , odd girth g and smallesteigenvalue θ min . In [6], Qiao and Koolen showed that for any fixed integer D ≥
3, there areonly finitely many such graphs with θ min ≤ − αk , where 0 < α < θ min compared with k . In the next result, we will show that if θ min is relatively close to − k ,then the odd girth g must be large. ∗ Corresponding authorE-mail addresses: [email protected] (Z. Qiao), [email protected] (Y. Jing),[email protected] (J. Koolen) heorem 1.1. Let Γ be a non-bipartite distance-regular graph with valency k and oddgirth g , having smallest eigenvalue θ min . Then there exists a constant ε ( g ) > such that θ min ≥ − (1 − ε ( g )) k . Remark 1.2.
The positive constant ε ( g ) goes to as the odd girth g goes to ∞ . Forexample, the (2 t + 1) -gon has valency k = 2 , odd girth g = 2 t + 1 and smallest eigenvalue θ min = 2 cos( tπ t +1 ) . Thus, ε ( g ) ≤ θ min k = 2 cos ( tπ t +1 ) . In [6], Qiao and Koolen classified non-bipartite distance-regular graphs with valency k ,diameter D ≤ θ min ≤ − k/
2. Using Theorem 1.1, we will classifynon-bipartite distance-regular graphs with valency k , diameter D and smallest eigenvalue θ min ≤ − D − D k , when D = 4 or 5. Theorem 1.3.
Let Γ be a non-bipartite distance-regular graph with valency k , diameter D and smallest eigenvalue θ min ≤ − D − D k .i) If D = 4 , then Γ is one of the following grapha) the Coxeter graph with intersection array { , , ,
1; 1 , , , } ,b) the 9-gon with intersection array { , , ,
1; 1 , , , } ,c) the Odd graph O with intersection array { , , ,
3; 1 , , , } ,d) the folded -cube with intersection array { , , ,
6; 1 , , , } .ii) If D = 5 , then Γ is one of the following grapha) the 11-gon with intersection array { , , , ,
1; 1 , , , , } ,b) the Odd graph O with intersection array { , , , ,
4; 1 , , , , } ,c) the folded -cube with intersection array { , , , ,
7; 1 , , , , } . This paper is organized as follows. In the next section, we give the definitions and somepreliminary results. In section 3, we give a proof of Theorem 1.1. In the last section, wegive a proof of Theorem 1.3.
For more background, see [4] and [7].All the graphs considered in this paper are finite, undirected and simple. Let Γ be agraph with vertex set V = V (Γ) and edge set E = E (Γ). Denote x ∼ y if the vertices x, y ∈ V are adjacent. The distance d ( x, y ) = d Γ ( x, y ) between two vertices x, y ∈ V (Γ)is the length of a shortest path connecting x and y . The maximum distance between twovertices in Γ is the diameter D = D (Γ). We use Γ i ( x ) for the set of vertices at distance i from x and write, for the sake of simplicity, Γ( x ) := Γ ( x ). The degree of x is the number | Γ( x ) | of vertices adjacent to it. A graph is regular with valency k if the degree of each of2ts vertices is k . The girth and odd girth of a graph is the length of its shortest cycle, andshortest odd cycle, respectively. A graph is called bipartite if it has no odd cycle.A connected graph Γ with diameter D is called distance-regular if there are integers b i , c i ( i = 0 , , . . . , D ) such that for any two vertices x, y ∈ V (Γ) with d ( x, y ) = i , thereare exactly c i neighbors of y in Γ i − ( x ) and b i neighbors of y in Γ i +1 ( x ), where we define b D = c = 0. In particular, Γ is a regular graph with valency k := b . We define a i := k − b i − c i ( i = 0 , , . . . , D ) for notational convenience. Note that a i = | Γ( y ) ∩ Γ i ( x ) | holds for any two vertices x, y with d ( x, y ) = i ( i = 0 , , . . . , D ).For a distance-regular graph Γ and a vertex x ∈ V (Γ), we denote k i := | Γ i ( x ) | and p hij := |{ w | w ∈ Γ i ( x ) ∩ Γ j ( y ) }| for any y ∈ Γ h ( x ). It is easy to seethat k i = b b · · · b i − / ( c c · · · c i ) and hence it does not depend on x . The numbers a i , b i and c i ( i = 0 , , . . . , D ) are called the intersection numbers , and the array { b , b , . . . , b D − ; c , c , . . . , c D } is called the intersection array of Γ. The matrix L is calledthe intersection matrix of Γ, where L = a b c a b c a ·· · · · · b D − c D a D . Let Γ be a distance-regular graph with v vertices and diameter D . Let A i ( i =0 , , . . . , D ) be the (0 , x, y )-entry is 1 whenever d ( x, y ) = i and 0 otherwise. We call A i the distance- i matrix and A := A the adjacency matrix of Γ. The eigenvalues θ > θ > · · · > θ D of thegraph Γ are just the eigenvalues of its adjacency matrix A . We denote m i the multiplicity of θ i . Note that the D + 1 distinct eigenvalues of Γ are precisely the eigenvalues of L (see[7, Proposition 2.7]).For each eigenvalue θ i of Γ, let U i be a matrix with its columns forming an orthonormalbasis for the eigenspace associated with θ i . And E i := U i U Ti is called the minimalidempotent associated with θ i , satisfying E i E j = δ ij E j and AE i = θ i E i , where δ ij is theKronecker delta. Note that vE is the all-ones matrix J .The set of distance matrices { A = I, A , A , . . . , A D } forms a basis of a commutative R -algebra A , known as the Bose-Mesner algebra . The set of minimal idempotents { E = v J, E , E , . . . , E D } is another basis of A . There exist ( D + 1) × ( D + 1) matrices P and Q (see [4, p.45]), such that the following relations hold A i = D X j =0 P ji E j and E i = 1 v D X j =0 Q ji A j ( i = 0 , , . . . , D ) . (1)Note that Q i = m i (see [4, Lemma 2.2.1]).Let E i = U i U Ti be the minimal idempotent associated with θ i , where the columns of U i form an orthonormal basis of the eigenspace associated with θ i . We denote the x -th row3f p v/m i U i by ˆ x . Note that E i ◦ A j = v Q ji A j , hence all the vectors ˆ x are unit vectorsand the cosine of the angle between two vectors ˆ x and ˆ y is u j ( θ i ) := Q ji Q i , where d ( x, y ) = j .The map x ˆ x is called a normalized representation and the sequence ( u j ( θ i )) Dj =0 is calledthe standard sequence of Γ, associated with θ i . As AU i = θ i U i , we have θ i ˆ x = P y ∼ x ˆ y , andhence the following holds: ( c j u j − ( θ i ) + a j u j ( θ i ) + b j u j +1 ( θ i ) = θ i u j ( θ i ) ( j = 1 , , . . . , D − ,c D u D − ( θ i ) + a D u D ( θ i ) = θ i u D ( θ i ) , (2)with u ( θ i ) = 1 and u ( θ i ) = θ i k . Lemma 2.1. (c.f. [7, Theorem 2.8])
Let Γ be a distance-regular graph with diameter D and v vertices. Let θ be an eigenvalue of Γ and ( u i ) Di =0 be the standard sequence associatedwith θ . Then the multiplicity m ( θ ) of θ as an eigenvalue of Γ satisfies m ( θ ) = v P Di =0 k i u i , (3) ≤ max { u , . . . , u j − , P ji =0 k i k j u j } ( j = 1 , , . . . , D ) . (4) Proof.
We only give a proof of Equation (4). v P Di =0 k i u i = P Di =0 k i P Di =0 k i u i ≤ max { P j − i =0 k i P j − i =0 k i u i , P Di = j k i P Di = j k i u i } , P j − i =0 k i P j − i =0 k i u i ≤ max { u , . . . , u j − } , P Di = j k i P Di = j k i u i ≤ P Di = j k i k i u i . Lemma 2.2. [4, Proposition 4.1.6]
Let Γ be a distance-regular graph with valency k anddiameter D . Then the following conditions holdi) c ≤ c ≤ · · · ≤ c D ,ii) k = b ≥ b ≥ · · · ≥ b D − ,iii) k i ’s ( i = 1 , , . . . , D ) are positive integers,iv) the multiplicities are positive integers. emma 2.3. (c.f. [2, Proposition 3.1]) Let Γ be a non-bipartite distance-regular graph withvalency k and odd girth g = 2 t + 1 . Then t X i =0 p i ( η ) u i ≥ , (5) where ( u i ) Di =0 is the standard sequence associated with the smallest eigenvalue θ min , η is anyeigenvalue of the g -gon, and p i ( x ) is defined as the following p ( x ) = 1 ,p ( x ) = x,p ( x ) = x − ,p i ( x ) = xp i − ( x ) − p i − ( x ) ( i = 3 , , . . . , t ) . (6) Proof.
Let ∆ be any g -gon in Γ. Let B i ( i = 0 , , . . . , t ) be the matrix with rows andcolumns indexed by V (∆), where the ( v, w )-entry is 1 whenever d Γ ( v, w ) = i and 0otherwise. Note that d ∆ ( v, w ) = d Γ ( v, w ) for any two vertices v, w ∈ V (∆), and wehave B i = p i ( B ) with p i ( x ) as Equation (6). By [2, Proposition 3.1], for any eigenvalue η of ∆, we have Equation (5). Lemma 2.4. (c.f. [6, Lemma 5.2])
Let Γ be a distance-regular graph with valency k andsmallest eigenvalue θ min . If a = 0 and θ min < − k , then c ≤ .Proof. Choose two vertices x, y ∈ V (Γ) with d ( x, y ) = 2. As a = 0, the subgraph inducedon x, y ∪ Γ( x ) ∪ Γ( y ) is a K ,c . Let x ˆ x be a normalized representation associated with θ = θ min . Consider the Gram matrix of the image of the K ,c with the bipartition, we seethat Q = (cid:18) (1 + u ) u u c (1 + ( c − u ) (cid:19) is positive semidefinite, by [4, Proposition 3.7.1 (iii)]. Then (1 , Q (1 , t ≥
0, which in turnimplies ( u + u )((2 + c ) − u u + u + 4 c ) ≥
0. As a = 0, we see u + u = ( θ + k )( θ − k ( k − <
0, thatis c c ≤ − − u u + u = k − θ − θ . When k >
1, we have θ < − k < − k and c ≤ k − θ − θ − k < Lemma 2.5.
Let Γ be a distance-regular graph with valency k and diameter D , havingsmallest eigenvalue θ min with associated standard sequence ( u i ) Di =0 . Then | u i +1 | ≥ | ( θ min − a i ) u i | − c i | u i − | b i ( i = 0 , , . . . , D − . (7) Proof.
By Equation (2) we see that u i +1 = ( θ min − a i ) u i − c i u i − b i ( i = 1 , , . . . , D ). As θ min < u i +1 , − u i and u i − has thesame sign. The result follows. 5 Main Theorem
In this section we will prove our main result.
Proof of Theorem 1.1 If g = 3, then θ min ≥ − k by [7, Proposition 2.11]. So we may assume g ≥
5. Let t = g − and ∆ be a g -gon in Γ. Let ( u i ) Di =0 be the standard sequence associated with the smallesteigenvalue θ = θ min .Assume c t ≤ ζ k for some ζ ≤ . By Lemma 2.3, we have P ti =0 p i ( η ) u i ≥
0, where p i ( x )is as Equation (6).We claim that there exist constants N i such that | u i − ( θk ) i | ≤ N i ζ ( i = 0 , , . . . , D ) . (8)Note that u = 1 and u = θk . Assume | u i − ( θk ) i | ≤ N i ζ for some 1 ≤ i ≤ t −
1. As c t ≤ ζ k ,we see that c i ≤ c t ≤ ζ k and b i = 1 − c i ≥ (1 − ζ ) k . Then | u i +1 − ( θk ) i +1 | = | θu i − c i u i − b i − ( θk ) i +1 |≤ | θb i u i − θb i ( θk ) i | + | θb i ( θk ) i − ( θk ) i +1 | + c i b i | u i − | = | θb i | · | u i − ( θk ) i | + c i b i · | ( θk ) i +1 | + c i b i · | u i − |≤ | kb i | N i ζ + c i b i + c i b i ≤ (2 N i + 4) ζ , where kb i ≤ − ζ ≤ c i b i ≤ ζ − ζ ≤ ζ ( ζ ≥ ). So we may take N = N = 0 and N i = 2 N i − + 4 ( i = 2 , , . . . , t ).Note that p i ( η ) is an eigenvalue of the distance- i graph of ∆. Hence | p i ( η ) | ≤ i = 0 , , . . . , t ), and by Equation (8), we have t X i =0 p i ( η ) u i ≤ t X i =0 p i ( η )( θk ) i + t X i =0 | p i ( η ) | · | u i − ( θk ) i |≤ t X i =0 p i ( η )( θk ) i + M ζ , (9)where M = P ti =0 N i .By Equation (6), we see that p i ( x ) = λ i + λ i ( i = 1 , , . . . , t ), with λ i = ( x ± √ x − i = 1 , f ( x, y ) = P ti =0 p i ( x ) y i = − ( λ y ) t +1 − λ y + − ( λ y ) t +1 − λ y −
1. Note that theeigenvalues of ∆ are 2 cos πjg ( i = 0 , , . . . , g − η = 2 cos π ( t − g , then we see f ( η, −
1) = − M , (10)6here M = 1 / cos ( t − πg . In fact, f (2 cos 2 πjg , −
1) = 1 − (cid:0) − e π i · jg (cid:1) t +1 − (cid:0) − e π i · jg (cid:1) + 1 − (cid:0) − e − π i · jg (cid:1) t +1 − (cid:0) − e − π i · jg (cid:1) −
1= ( − t · e π i · j ( t +1) g + e − π i · jtg e π i · jg = ( − t + j / cos jπg . Take ζ = min { M M , } . Note that M , M , and hence ζ is determined by g . By Equation(10), we see f ( η, −
1) + M ζ ≤ − M <
0. We also have f ( η,
0) + M ζ = 1 + M ζ >
0. ByEquation (5) and (9), we have 0 ≤ f ( η, θk ) + M ζ . Take − (1 − ε ( ζ )) as the smallest root y of the equation f ( η, y ) + M ζ = 0 in the interval ( − , θ ≥ − (1 − ε ( ζ )) k .Now we consider the case c t > ζ k .If c t >
1, then we claim that the diameter D ≤ tζ and θ min ≥ − (1 − ε ( ζ )) k for someconstant ε ( ζ ) >
0. Without loss of generality, we may assume 4 i t ≤ D ≤ i +1 t for someinteger i ≥
1. If c t − j = c t − = 1, by [7, Theorem 7.1], we see j ≤ t −
1, that is c t − > c t − . Then c t − = c t + j > c t − j for some 0 ≤ j ≤ t −
1, and c t − ≥ c t by [7, Proposition 7.2]. This implies k ≥ c i t ≥ i c t , that is D ≤ t ( kc t ) ≤ tζ . Then by[6, Theorem 1.1], the set S of distance-regular graphs with valency k , diameter D ≤ tζ ,smallest eigenvalue θ min ≤ − (1 − ε ( ζ )) k and odd girth g is finite. Take ε ( ζ ) = min Γ ∈ S k + θ min k , if S = ∅ ,ε ( ζ ) , otherwise . If c t = 1, then k < ζ . The set S ′ of distance-regular graphs with valency k < ζ andodd girth g is finite, by [1, Theorem 1.1]. Take ε ( ζ ) = min Γ ∈ S ′ k + θ min k , if S ′ = ∅ ,ε ( ζ ) , otherwise . Take ε = min { ε ( ζ ) , ε ( ζ ) , ε ( ζ ) } and the result follows. Remark 3.1.
When the odd girth g = 5 and c ≤ ζ k , we may take N = − ζ . Then f ( x, y )+ M ζ = 1+ xy +( x − y + ζ − ζ . By substituting η = 2 cos π into f ( η, θk )+ M ζ ≥ ,we find an inequality between ζ and θk . For example, if ζ = 0 . , then θ ≥ − . k . θ minIn this section we study distance-regular graphs with relatively small θ min . In the restof this section we will give a proof of Theorem 1.3.7 roof of Theorem 1.3 Assume Γ has odd girth g = 2 t + 1. Let ( u i ) Di =0 be the standard sequence associated withthe smallest eigenvalue θ = θ min .We first consider the case D = 4. We may assume k ≥
5, otherwise Γ is the 9-gon orthe Coxeter graph by [3] and [5, Theorem 1.1].As θ < − k , by [7, Proposition 2.11], we have a = 0. If a = 0, that is t = 2, thensubstitute η = 2 cos π ( t − g into Equation (5) and we get ( k − t )(2 k + √ t + t + √ − k ( k − ≥
0, whichimplies that θ ≥ − k −√ √ . Combine it with θ ≤ − k , we see that k ≤
2. Hence a = 0.Note that θ ≤ − k < − k , by Lemma 2.4, we see c ≤ a = 0, then consider t X i =0 p i ( η ) u i ≥ − D − D k ≥ θ (11)with η = 2, we obtain that k ≤ c = 1, and k ≤ c = 2. No intersection arrayssatisfy Lemma 2.2, with 5 ≤ k ≤ D = 4, a = a = 0 = a , c = 2 and θ min ≤ − k .Hence a = 0.Assume k ≥
36. Since k ≥ c ≤ θ ≤ − k , by Equation (7), we obtain | u | ≥ . | u | ≥ . L of Γ, where L = k k − c k − c
00 0 c k − c c k − c . We see that k + θ ≤ tr ( L ) ≤ k + 6 k + c (2 k − c ), where c ≤ c ≤ c .Since k ≥
36 and θ ≤ − k , we obtain that c k ≥ . m ≤ max { u , u , k + k k u } . Since k b = k c , we see k + k k u ≤ u (1 + kc ). With | u | ≥ , | u | ≥ . | u | ≥ . c k ≥ . m <
36. By [4, Theorem 4.4.4],we see that k ≤ m <
36, a contradiction. It follows that k ≤
35. Then we check theintersection arrays satisfy Lemma 2.2, with 5 ≤ k ≤ D = 4, a = a = a = 0 = a , c = 1 or 2 and θ min ≤ − k , and we get the folded 9-cube and odd graph O . This showsthe case D = 4.Now we consider the case D = 5. Similar to the case D = 4, we may assume k ≥ θ < − k , by [7, Proposition2.11], we have a = 0. Substitute η = 2 cos π ( t − g with t = 2 into Equation (5), weobtain θ ≥ − k −√ √ . Together with θ ≤ − k , we see k ≤
2, and hence a = 0. Since θ ≤ − k < − k , by Lemma 2.4, we have and c ≤ a = 0, then consider Equation (11) with η = 2, we obtain that k ≤ c = 1, and k ≤ c = 2. By [4, Theorem 1.13.2], no such graphs exist with k = 5 and c = 2.Hence a = 0. 8e consider a = 0. If c ≤ . k , combine it with Equation (11), where η = − g = 9), we see that k ≤
24. Assume k ≥
24, then c ≥ . k . By Equation (7), weobtain | u | ≥ . | u | ≥ . k k = b c ≤ − c c and k k = b b c c ≤ ( − c c ) . ByLemma 2.1, we see that m ≤ max { u , u , u (1 + 1 − c c + ( 1 − c c ) ) } , (12)that is k ≤ m ≤
24 (Lemma [4, Theorem 4.4.4]). No intersection arrays satisfy Lemma 2.2with 5 ≤ k ≤ D = 5, a = a = a = 0 = a , c = 1 or 2 and θ ≤ − k . Hence a = 0.Assume k ≥
71. Then by Equation (7), we see that | u | ≥ . | u | ≥ . θ ≤ − k , c = 1 or 2. Then as m ≥ k ≥
71, by Equation (12), we obtain c ≤ . k . It implies | u | ≥ . L , and we see that k + θ ≤ tr ( L ) ≤ k + 6 k + 4 c k − c , which implies c ≥ . k . Andwe see k ≤ m ≤ min { u , u , u , u (1 + kc ) } ≤
71. It follows that k ≤
71. Then we check allintersection arrays satisfy Lemma 2.2 with 5 ≤ k ≤ D = 5, a = a = a = a = 0 = a , c = 1 or 2 and θ ≤ − k and we obtain the odd graph O and the folded 11-cube. Thisshows the case D = 5. Acknowledgments
JHK is partially supported by the National Natural Science Foundation of China (GrantNo. 11471009 and Grant No. 11671376) and by ’Anhui Initiative in Quantum InformationTechnologies’ (Grant No. AHY150200). ZQ is partially supported by the National NaturalScience Foundation of China (Grant No. 11801388).
References [1] S. Bang, A. Dubickas, J. Koolen, and V. Moulton. There are only finitely manydistance-regular graphs of fixed valency greater than two.
Adv. Math. , 269:1–55, 2015.[2] S. Bang, J. Koolen, and J. Park. Some results on the eigenvalues of distance-regulargraphs.
Graphs and Combin. , 31:1841–1853, 2015.[3] N. Biggs, A. Boshier, and J. Shawe-Taylor. Cubic distance-regular graphs.
J. LondonMath. Soc. (2) , 33:385–394, 1986.[4] A. Brouwer, A. Cohen, and A. Neumaier.
Distance-Regular Graphs . Springer-Verlag,1989.[5] A. Brouwer and J. Koolen. The distance-regular graphs of valency four.
J. AlgebraicCombin. , 10:5–24, 1999.[6] Z. Qiao and J. Koolen. A valency bound for distance-regular graphs.
J. Combin. TheorySer. A , 155:304–320, 2018. 97] E. van Dam, J. Koolen, and H. Tanaka. Distance-regular graphs.