Distance-regular graphs with diameter 3 and eigenvalue a2 − c3

Journal Pre-proof Distance-regular graphs with diameter 3 and eigenvalue a2 − c3

Quaid Iqbal, Jack H. Koolen, Jongyook Park, Masood Ur Rehman

PII:

S0024-3795(19)30461-6

DOI:

https://doi.org/10.1016/j.laa.2019.10.021

Reference:

LAA 15158

To appear in:

Linear Algebra and its Applications

Received date:

19 December 2018

Accepted date:

23 October 2019

Please cite this article as: Q. Iqbal et al., Distance-regular graphs with diameter 3 and eigenvalue a2 − c3 , Linear Algebra Appl. (2020), doi: https://doi.org/10.1016/j.laa.2019.10.021.

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier.

DISTANCE-REGULAR GRAPHS WITH DIAMETER 3 AND EIGENVALUE a2 − c3 QUAID IQBAL† , JACK H. KOOLEN†§ , JONGYOOK PARK∗ , MASOOD UR REHMAN† † SCHOOL

OF MATHEMATICAL SCIENCES, UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA, 96 JINZHAI ROAD, HEFEI, 230026, ANHUI, PR CHINA § WEN-TSUN WU KEY LABORATORY OF CAS, 96 JINZHAI ROAD, HEFEI, 230026, ANHUI, PR CHINA ∗ DEPARTMENT OF MATHEMATICS, KYUNGPOOK NATIONAL UNIVERSITY, DAEGU, 41566, REPUBLIC OF KOREA E-MAIL: [email protected], [email protected], [email protected], [email protected] Abstract. In this paper, we consider the distance-regular graphs Γ whose distance-2 graphs Γ2 are strongly regular. Note that if Γ is bipartite, then its distance-2 graph is not connected. We first show that the distance-2 graph of a non-bipartite distanceregular graph with diameter 3 and eigenvalue a2 −c3 is strongly regular, and then we give several kinds of classifications of non-bipartite distance-regular graphs with diameter 3 and eigenvalue a2 − c3 .

Keywords : distance-regular graphs, strongly regular graphs, distance-i graphs, eigenvalues, geometric AMS classification 05C50, 05E30

1. Introduction In this paper, we consider the distance-regular graphs Γ whose distance-2 graphs Γ2 are strongly regular. We note that if Γ is bipartite, then its distance-2 graph is not connected. So, we are interested in the class of non-bipartite distance-regular graphs. This extends a paper by Brouwer [4] in 1984. He studied distance-regular graphs with diameter 3 such that their distance-2 graphs are strongly regular graphs. For example, he showed that for an antipodal non-bipartite distance-regular graph Γ with diameter 3, its distance-2 graph Γ2 is strongly regular if, and only if, the graph Γ has intersection array {st, s(t − 1), 1; 1, t − 1, st} for some integers s, t ≥ 2. And in this case, the complement of Γ2 , that is the distance-{1, 3} graph of Γ, is a strongly regular graph with parameters ((st + 1)(s + 1), s(t + 1), s − 1, t + 1). Note that a strongly regular graph with parameters 1

((st + 1)(s + 1), s(t + 1), s − 1, t + 1) for positive integers s, t, is called pseudo-geometric. In this paper, we will extend this result. Assume that Γ is a non-bipartite distance-regular graph with diameter D. It is known that the adjacency matrix of Γ2 is obtained from the adjacency matrix of Γ and the intersection numbers of Γ (see, for example [2, p.127]). That is, we could find the eigenvalues of Γ2 from the eigenvalues of Γ. This means that Γ2 has at most D +1 distinct eigenvalues as Γ has exactly D + 1 distinct eigenvalues. Also, it is known that if Γ has 2 pairs of distinct eigenvalues θ and θ of Γ satisfying θ + θ = a1 , then Γ2 has exactly D + 1 −  distinct eigenvalues. We note that for D = 3, there exist distinct eigenvalues θ and θ of Γ satisfying θ + θ = a1 if, and only if, Γ has an eigenvalue a2 − c3 . This means that the distance-2 graph of a non-bipartite distance-regular graph with diameter 3 and eigenvalue a2 − c3 is strongly regular. We give several kinds of classifications of non-bipartite distance-regular graphs with diameter 3 and eigenvalue a2 − c3 under various conditions, for example the valency k is at most 2(a1 + 1), c3 ≤ 9, a2 ≤ 7 and so on. There are several related studies. In [4] Brouwer also studied distance-regular graphs with diameter three whose distance-3 graph is strongly regular. In 2017, Bang and Koolen [1] extended this study. They studied distance-regular graphs Γ with diameter 3 having eigenvalue −1, where the statement that Γ has an eigenvalue −1 is equivalent to the fact that the distance-3 graph of Γ is strongly regular. In 2016, Dalf´o, Fiol and Koolen [6] studied the spectral excess theorem for graphs with a few distinct eigenvalues whose distance-2 or distance-1-or-2 graph is strongly regular. In 2018, Brouwer, Cioab˘a, Ihringer and McGinnis [3] studied the smallest eigenvalues of the distance-i graphs for several families of distance-regular graphs. In the following section, we introduce the preliminaries, and we give properties of distance-regular graphs with diameter 3 and eigenvalue a2 − c3 in Section 3. Then in Section 4, we give several kinds of classifications of distance-regular graphs with diameter 3 and eigenvalue a2 − c3 . 2. Preliminaries All the graphs considered in this paper are finite, undirected and simple. The reader is referred to [2] for more information. Let Γ be a connected graph with vertex set V (Γ). The distance dΓ (x, y) between two vertices x, y ∈ V (Γ) is the length of a shortest path between x and y in Γ. The diameter D = D(Γ) of Γ is the maximum distance between any two vertices of Γ. For each x ∈ V (Γ), let Γi (x) be the set of vertices in Γ at distance i from x (0 ≤ i ≤ D). In addition, define Γ−1 (x) = ∅ and ΓD+1 (x) = ∅. For the sake of simplicity, let Γ(x) = Γ1 (x) and we denote x ∼ y if two vertices x and y are adjacent. For a vertex x of Γ, the number |Γ(x)| is called the valency of x in Γ. In particular, Γ is regular with valency k if k = |Γ(x)| holds for all x ∈ V (Γ). Let Γ be a graph with diameter D and let x, y be vertices of Γ at distance i (0 ≤ i ≤ D). Then the number of vertices which are at distance j from x and h from y is 2

denoted by pijh (x, y) and is called an intersection number of Γ. Note that pijh (x, y) = |Γj (x) ∩ Γh (y)|. And we consider the numbers ci (x, y) = pii−1,1 (x, y), ai (x, y) = pii1 (x, y), bi (x, y) = pii+1,1 (x, y). Note that |Γ(y)| = ci (x, y) + ai (x, y) + bi (x, y) holds for all 0 ≤ i ≤ D. The intersection numbers pijh (x, y) (0 ≤ i, j, h ≤ D) are called well-defined if these numbers do not depend on the choice of x and y but only on i, i.e., pijh (x, y) = pijh (z, w) if dΓ (x, y) = dΓ (z, w) = i. If this is the case, then these numbers are denoted simply by pijh (0 ≤ i, j, h ≤ D). A connected graph Γ with diameter D is called distance-regular if the numbers ci (x, y), ai (x, y) and bi (x, y) are well-defined for 0 ≤ i ≤ D. For distance-regular graphs, these numbers are denoted simply by ci , ai and bi for 0 ≤ i ≤ D. Also, for distance-regular graphs and a given vertex x, the numbers |Γi (x)| (0 ≤ i ≤ D) are simply denoted by ki . The array {b0 = k, b1 , . . . , bD−1 ; c1 = 1, c2 , . . . , cD } of intersection numbers of a distance-regular graph Γ with valency k and diameter D is called the intersection array of Γ, and a distance-regular graph with intersection array {k, μ, 1; 1, μ, k} is called a Taylor graph. A clique in a graph is a set of pairwise adjacent vertices. A complete graph is a graph whose vertex set is a clique. The adjacency matrix A = A(Γ) of a graph Γ is the matrix whose rows and columns are indexed by V (Γ), where the (x, y)-entry is 1 whenever x ∼ y and 0 otherwise. The eigenvalues of Γ are the eigenvalues of A. The multiset of eigenvalues of a graph Γ is called the spectrum of Γ. For a distance-regular graph Γ with valency k, diameter D ≥ 2 and the smallest eigenvalue θD , it is known that the size of a clique C in Γ is bounded by |C| ≤ 1 − k/θD ([2, Proposition 4.4.6]). A clique C in Γ is called a Delsarte clique if C contains exactly 1 − k/θD vertices. A distance-regular graph Γ with diameter D ≥ 2 is called geometric if there exists a set C of Delsarte cliques such that each edge of Γ lies in a unique C ∈ C.   ˜Δ , c˜Δ , A non complete graph Δ is called a strongly regular graph with parameters v˜Δ , k˜Δ , a if it is a regular graph with valency k˜Δ and any two adjacent (nonadjacent, respectively) cΔ , respectively) common neighbors, where vertices have exactly a ˜Δ (˜   v˜Δ := |V (Δ)|. We ¯ ¯ denote the parameters of the complement Δ of Δ by v¯Δ , kΔ , a ¯Δ , c¯Δ . If there is no con    ˜ ˜Δ , c˜Δ and v¯Δ , k¯Δ , a ¯Δ , c¯Δ . We note that fusion, then we omit Δ from notation v˜Δ , kΔ , a a distance-regular graph with diameter 2 is a strongly regular graph. Let Γ be a connected graph with diameter D. For a subset S of {1, 2, . . . , D}, the distance-S graph ΓS of Γ is the graph with vertex set V (Γ) and whose vertices are adjacent if they are at distance i ∈ S from each other. If S is a singleton, say S = {i}, then the distance-S graph is also called distance-i graph and simply denoted by Γi . A graph Γ is called bipartite if it has no odd cycle. (If Γ is a distance-regular graph with diameter D and bipartite, then a1 = a2 = · · · = aD = 0.) An antipodal graph is a connected graph Γ with diameter D > 1, for which its distance-D graph ΓD is a disjoint union of complete graphs. Moreover, if all complete graphs have the same size r, then Γ is also called an antipodal r-cover. Note that a Taylor graph is an antipodal 2-cover. We call a distance-regular graph Γ primitive if it is connected and all Γi (i = 1, · · · , D) are 3

connected, and imprimitive otherwise. Here, we note that an imprimitive distance-regular graph with valency k > 2 is bipartite or antipodal (or both) (see [2, Theorem 4.2.1]). Let Γ be a distance-regular graph with diameter D. It is known that Γ has exactly D + 1 distinct eigenvalues θ0 > θ1 > ... > θD which are the eigenvalues of the tridiagonal (D + 1) × (D + 1)-matrix ⎡

0 k ⎢ 1 a1 b1 0 ⎢ ⎢ b2 c 2 a2 L1 (Γ) = ⎢ .. .. .. ⎢ . . . ⎢ ⎣ 0 cD−1 aD−1 bD−1 cD aD

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(cf. [2, p. 129]). Let mi := mθi denote the multiplicity of the eigenvalue θi (i = 1, . . . , D). It is known that for each integer j ≥ 0, D

mi θij = tr(Aj ) = |{w | w is a closed walk of length j in Γ}|

i=0

holds, where tr(Aj ) is the trace of matrix Aj . The following two theorems, due to Neumaier [11], are on strongly regular graphs which will be used later. Theorem 2.1. ([11, Theorem 3.1]) For a nontrivial strongly regular graph Δ with pa˜ a rameters (˜ v , k, ˜, c˜) and integral eigenvalues k˜ > n − m > −m satisfying 1 < m < n, we have c˜ ≤ m3 (2m − 3). Equality implies n = m(m − 1)(2m − 1). Theorem 2.2. ([11, Theorem 4.7]) Let Δ be a strongly regular graph with parameters ˜ a (˜ v , k, ˜, c˜) and integral eigenvalues k˜ > σ > τ = −m, where m > 1. Then at least one of the following hold: (i) The intersection number c˜ satisfies c˜ = m(m − 1), and Δ is a Latin Square graph; (ii) The intersection number c˜ satisfies c˜ = m2 , and Δ is a Steiner graph; (iii) Suppose that c˜ ∈ / {m(m − 1), m2 }. Then the eigenvalue σ satisfies σ ≤ 12 m(m − 1)(˜ c + 1) − 1. 2.1. A construction of Brouwer. A generalized quadrangle is a partial linear space (P, L), such that for any non-incident point-line pair (p, L) there exists exactly one point on L collinear with p. The generalized quadrangle is of order (s, t), if all lines have s + 1 points, and there are exactly t + 1 lines through any point. (For short, we write GQ(s, t), instead of a generalized quadrangle of order (s, t).) By interchanging the role of P and 4

L of a GQ(s, t) we obtain a GQ(t, s). A standard reference on the subject of generalized quadrangles is Payne and Thas [12]. The point graph of a GQ(s, t) is a strongly regular graph with parameters ((st + 1)(s + 1), s(t + 1), s − 1, t + 1). We will also denote the point graph of a GQ(s, t) by GQ(s, t). A strongly regular graph with parameters ((st + 1)(s + 1), s(t + 1), s − 1, t + 1) (for some positive integers s and t) is called pseudo-geometric and we denote such a strongly regular graph by pseudoGQ(s, t). It is not difficult to see that a pseudoGQ(s, t) is geometric if and only if every edge is contained in a clique of order s + 1. The following lemma gives necessary conditions for the existence of a pseudoGQ(s, t), (cf. [7, p. 235-236]). Lemma 2.3. Let Γ be a strongly regular graph with parameters ((st + 1)(s + 1), s(t + 1), s − 1, t + 1), with s, t positive integers. Then the following hold: (1) If s ≥ 2, then 1 ≤ t ≤ s2 holds. is a positive integer. (2) st(s+1)(t+1) s+t (3) Moreover, if Γ is geometric and t ≥ 2, then 1 ≤ s ≤ t2 holds. A spread R of a pseudoGQ(s, t) Γ is a set of cliques of Γ of order s + 1, such that each vertex lies in exactly one clique of R. Let Γ be a pseudoGQ(s, t) with spread R. Define a graph Σ = Σ(Γ, R) with vertex set V (Γ) and {x, y} is an edge in Σ, if {x, y} is an edge of Γ but not an edge in any of the cliques of R, i.e., Σ is obtained from Γ by removing the edges of the spread R. The graph Σ is an antipodal distance-regular graph with diameter 3 and has intersection array {st, s(t − 1), 1; 1, t − 1, st} and, hence, Σ is an antipodal (s + 1)-cover of the complete graph Kst+1 . Note that for different spreads R1 and R2 in Γ, the graphs Σ(Γ, R1 ) and Σ(Γ, R2 ) may be non-isomorphic. The following result has been shown by Brouwer [4], and it also follows from Proposition 3.4 by setting b2 = 1. Proposition 2.4. Let Γ be an antipodal non-bipartite distance-regular graph with diameter 3. Then the distance-2 graph Γ2 of Γ is strongly regular if, and only if, Γ has intersection array {st, s(t − 1), 1; 1, t − 1, st} with s, t ≥ 2 integers. In this case, the complement of Γ2 is a pseudoGQ(s, t). Remark 2.5. For a positive integer n, there exists a GQ(2n , 2n ) with a spread, and hence there exists an antipodal distance-regular graph with intersection array {22n , 2n (2n − 1), 1; 1, 2n − 1, 22n }, see Payne and Thas [12]. In 2012, Koolen and Park showed the following classification of distance-regular graphs with valency k, and diameter D ≥ 3 satisfying k ≤ 2(a1 + 1). Proposition 2.6. ([9, Theorem 16]) Let Γ be a distance-regular graph with valency k and diameter D ≥ 3. If k ≤ 2(a1 + 1), then Γ is one of the following: (i) A polygon; (ii) The line graph of a Moore graph of diameter 2; 5

(iii) (iv) (v) (vi)

A regular generalized 2D-gon of order (s, 1) with D ∈ {3, 4, 6}; A Taylor graph; The Johnson graph J(7, 3); The halved 7-cube.

3. Properties of distance-regular graphs with diameter 3 and eigenvalue a2 − c3 In this section, we will find properties of distance-regular graphs with diameter 3 and eigenvalue a2 −c3 . In order to do so, we first recall the following theory of distance-regular graphs [2]. Let Γ be a distance-regular graph with diameter D and Γ2 be its distance-2 graph. Then the adjacency matrix A2 of Γ2 can be expressed as (A2 − a1 A − kI) , c2 where A is the adjacency matrix of Γ. This shows that if θ is an eigenvalue of Γ with 2 eigenvector v then f (θ) = (θ −ac21 θ−k) is an eigenvalue of Γ2 with eigenvector v. Note that Γ has exactly D +1 distinct eigenvalues. We consider two distinct eigenvalues of Γ, say θ and θ . Then f (θ) = f (θ ) if, and only if, θ + θ = a1 . Let L(Γ) be the set of distinct eigenvalues θ of Γ such that there exists an eigenvalue θ of Γ distinct from θ satisfying θ+θ = a1 , with |L(Γ)| := 2. Then Γ2 has exactly D+1− distinct eigenvalues. And we note that Γ2 is connected if, and only if Γ is not bipartite. By applying this to distance-regular graphs with diameter 3 we obtain the following results. A2 =

Lemma 3.1. Let Γ be a non-bipartite distance-regular graph with diameter 3 and distinct eigenvalues k = θ0 > θ1 > θ2 > θ3 , then the following are equivalent. (i) a2 − c3 is an eigenvalue of Γ; (ii) There are distinct eigenvalues θ and θ of Γ such that θ + θ = a1 holds; (iii) The distance-2 graph Γ2 of Γ is strongly regular; (iv) The distance-2 graph Γ2 of Γ is co-edge-regular with parameter c˜ = ac22b1 ; (v) c3 (a3 + a2 − a1 ) = a2 b1 ; (vi) (a2 − c3 )(b1 − c3 ) = c3 . If Γ satisfies (i)-(vi), then the following statements hold: (a) {a2 − c3 , θ, θ } = {θ1 , θ2 , θ3 }; / {0, −1, −c3 }; (b) a2 − c3 ∈ (c) c˜ − 1 ≥ a2 ≥ 1; (d) c3 /2 ≤ a2 ≤ 2c3 ; (e) c3 /2 ≤ b1 ≤ 2c3 . Proof. (i)⇔(ii)⇔(iii): By [7, Lemma 10.2.1], a connected regular graph Δ is strongly regular if, and only if, Δ has at most 3 distinct eigenvalues. We obtain that Γ2 has at 6

most 3 distinct eigenvalues if, and only if, there exist distinct eigenvalues θ and θ of Γ such that θ + θ = a1 . As k + θ1 + θ2 + θ3 = a1 + a2 + a3 , we find that Γ2 is strongly regular if, and only if, a2 − c3 is an eigenvalue of Γ. (iii)⇔(iv)⇔(v): As Γ is non-bipartite and distance-regular, the distance-2 graph Γ2 of ˜ = p222 . Γ is connected, regular with valency k˜ = k2 and edge-regular with parameter a This shows that the graph Γ2 is strongly regular if, and only if, Γ2 is co-edge regular with 2 −a1 ) parameter c˜, where c˜ satisfies c˜ = p122 = p322 . As p122 = kk2 p221 = cb12 a2 and p322 = c3 (a3 +a , c2 we see that (iii), (iv) and (v) are equivalent. (v)⇔(vi): As a3 = k − c3 = (b1 + a1 + 1 − c3 ), we find that (vi) is equivalent to (v). Now we assume that (i)-(vi) all hold. Then by the proof of (i)⇔(ii)⇔(iii) we see that {a2 − c3 , θ, θ } = {θ1 , θ2 , θ3 } holds, and this shows (a). As Γ2 is connected and noncomplete, we see that c˜ is not zero, and this implies that a2 = 0. As b1 = 0, c3 = 0 and / {0, −1, −c3 }, and this shows (b). As Γ has diameter a2 = 0, we find, by (vi), that a2 −c3 ∈ 3, we know, by [2, Proposition 4.1.6], that b1 ≥ c2 . If b1 = c2 , then k = k2 and this shows, by [2, Proposition 5.1.1], that Γ is the 7-gon or an antipodal 2-cover. But both graphs do not have that Γ2 is strongly regular, and hence we may assume that b1 > c2 . As c˜ = ac22b1 and b1 > c2 we obtain c˜ > a2 , and this shows (c). (d) and (e) follow easily from (vi).  In the following proposition, we give properties of the eigenvalue a2 − c3 of a distanceregular graph with diameter 3. Proposition 3.2. Let Γ be a non-bipartite distance-regular graph with diameter 3 and distinct eigenvalues k = θ0 > θ1 > θ2 > θ3 . Let θ = a2 − c3 be an eigenvalue of Γ. Then one of the following holds: (i) c3 ≥ θ ≥ 1, θ = θ2 and θ1 , θ2 , θ3 are integers. (ii) −2 ≥ θ ≥ − c23 and θ = θ3 . Proof. The three eigenvalues θ1 , θ2 , θ3 are the three distinct eigenvalues of the matrix ⎤ ⎡ 0 −1 b1 ⎦ b2 T = ⎣ 1 a1 + 1 − c2 0 c2 a3 − b 2 (see [2, p. 130]). By removing the second column and second row of T , we obtain the following matrix  −1 0  . T = 0 a 3 − b2 Note that the matrix T  has eigenvalues −1 and a3 − b2 . As a3 + c3 = k > a2 + b2 , we find that θ = a2 − c3 < a3 − b2 . If θ < −1, then, by interlacing, we see that θ = θ3 . By Lemma 3.1, we find a2 ≥ c3 /2, and hence θ = a2 − c3 ≥ −c3 /2 holds. This shows (ii). 7

Now we assume that θ ≥ −1. By Lemma 3.1, we find θ ≥ 1 and, by interlacing, we find θ = θ2 , as in this case −1 < 1 ≤ θ < a3 − b2 holds. To complete the case (i), we still need to show that θ1 , θ2 , θ3 are integers in this case. Let m1 , m2 and m3 be multiplicities of eigenvalues θ1 , θ2 and θ3 respectively. By Lemma 3.1, we have θ1 + θ3 = a1 ≥ 0 and θ = θ2 ≥ 1, and hence m1 = m3 , as otherwise 0 = tr(A) = k + m1 θ1 + m2 θ2 + m3 θ3 = k + m2 θ + m1 a1 > 0 holds, which is impossible, where tr(A) is the trace of the adjacency matrix A of Γ. As θ2 = θ = a2 − c3 is integral and m1 = m3 , we know that θ1 and θ3 are integers. This shows (i).  In the following lemma, we will show that if a distance-regular graph with valency k and diameter 3 satisfies c3 = k, then its distance-3 graph is also a distance-regular graph with diameter 3. And then in Proposition 3.4, we will look at distance-regular graphs with valency k, diameter 3 and eigenvalue a2 − c3 satisfying c3 = k. The following lemma follows immediately from [2, Lemma 4.1.7], but we give a proof for the convenience of readers. Lemma 3.3. Let Γ be a primitive distance-regular graph with valency k and diameter 3 satisfying c3 = k. Then the distance-3 graph of Γ is also a distance-regular graph with intersection array {k3 , p323 , p213 ; 1, p233 , k3 }. Proof. Let x be a vertex of Γ. Note that x is also a vertex of Γ3 . Since there are k3 vertices at distance 3 from x in Γ, Γ3 has valency k3 . Let y be a neighbor of x in Γ3 , i.e., y is at distance 3 from x in Γ. We consider intersection numbers p303 = 1, p313 , p323 and p333 . As a3 = 0, we know that p313 = 0. Note that p323 = 0 otherwise Γ is antipodal. And p333 means the number of common neighbors of two adjacent vertices of Γ3 . This shows that y has p323 neighbors that are at distance 2 from x in Γ3 and those vertices are also at distance 2 from x in Γ. Note that each vertex at distance 2 from x in Γ3 has p233 common neighbors with x in Γ3 . As k3 p323 = k2 p233 , we know that there are k2 vertices at distance 2 from x in Γ3 , and this shows that there are k vertices at distance 3 from x in Γ3 . Note that each vertex at distance 2 from x in Γ3 has p213 neighbors that are at distance 3 from x in Γ3 . As k2 p213 = kp123 and p123 = b1c2b2 = k3 , we complete the proof.  Proposition 3.4. Let Γ be a non-bipartite distance-regular graph with valency k, diameter 3, and distinct eigenvalues k = θ0 > θ1 > θ2 > θ3 . Assume that θ = a2 −c3 is an eigenvalue of Γ. If c3 = k, then the following hold: (i) There exist integers s, t both at least 2, such that k = st, a1 = s − 1 and θ3 = −t = θ; (ii) The distance-{1, 3} graph Γ{1,3} of Γ is a pseudoGQ(s, τ ) where τ = t(t−1) ; c2 (iii) All the eigenvalues of Γ are integral. Proof. (i): As c3 = k and a2 < k, θ = a2 − c3 is negative. By Proposition 3.2, we know that θ3 = a2 − c3 = θ is an integer. Let θ3 = −t. Then we have a2 = c3 + θ3 = k − t. By Lemma 3.1, we find that a1 + 1 = k − b1 = c3 − b1 = kt . Let kt = s. We note that if s = 1, 8

then k = t implies a2 = k − t = 0, which contradicts Lemma 3.1 (b). Hence s ≥ 2 is an integer and a1 = s − 1. This shows (i). (ii): We note that the distance-2 graph Γ2 of Γ is a strongly regular graph and that the complement of Γ2 is the distance-{1, 3} graph Γ{1,3} of Γ, i.e., Γ{1,3} is also a strongly regular graph. By (i), we find that Γ has intersection array {st, s(t − 1), t − c2 ; 1, c2 , st}, 2 2) and this shows that Γ{1,3} has 1 + st + s t(t−1) + s(t−1)(t−c = (s + 1)( st(t−1) + 1) vertices c2 c2 c2 s(t−1)(t−c2 ) t(t−1) and valency st + = s( c2 + 1). As Γ{1,3} is a strongly regular graph, any two c2 adjacent vertices of Γ{1,3} has p333 = p111 = a1 = s − 1 common neighbors. And then we could find that any two non-adjacent vertices of Γ{1,3} have t(t−1) + 1 common vertices. c2 t(t−1) Let τ = c2 . Then Γ{1,3} is a strongly regular graph with parameters ((s + 1)(sτ + 1), s(τ + 1), s − 1, τ + 1). This shows (ii). (iii): If Γ is antipodal, then Γ has distinct eigenvalues k = st, s, −1 and −t (see, [2, p. 431]). So, we may assume that Γ is primitive. By Lemma 3.1, we know that θ1 +θ2 = a1 = s − 1. Note that the distance-{1, 3} graph Γ{1,3} of Γ has eigenvalues s(τ + 1), s − 1 and (τ +1) s2 (sτ +1) and (see, [7, Lemma 10.8.2]) −τ − 1 with respective multiplicities 1, s(s+1)τ s+τ s+τ and that Γ3 is also a distance-regular graph with diameter 3 by Lemma 3.3, and hence has the same primitive idempotents (see, [2, Chapter 4]). Let mi be the multiplicity of (τ +1) and θ1 m1 + θ2 m2 = θi for i = 0, 1, 2, 3. Then, by (ii), we have m1 + m2 = s(s+1)τ s+τ ts2 (sτ +1) s+τ

s(s+1)τ (τ +1) s+τ s(s+1)τ (τ +1)(s−1) = s+τ

− st. If θ1 is not integral, then m1 = m2 , i.e., 2m1 = m1 + m2 = ts2 (sτ +1) − st hold. This implies s+τ ts2 (sτ +1)−st(s+τ ) 2 = 2 stτ (s+1)(s−1) , and hence we s+τ s+τ

and (θ1 + θ2 )m1 = θ1 m1 + θ2 m2 =

have τ +1 = 2t. 2(s−1)m1 = 2(θ1 +θ2 )m1 = t(t−1) As τ = c2 , we find 2c2 t = t(t − 1) + c2 and that means t divides c2 . This is impossible,  as a2 = k − t implies c2 < t. This finishes (iii). Remark. The following arrays are feasible intersection arrays for distance-regular graphs with valency k, diameter 3 and eigenvalue a2 − c3 satisfying c3 = k. Note that the distance-3 graph of each of them would also be a distance-regular graph with diameter 3. (i) {210, 200, 9; 1, 12, 210} with spectrum {2101 , 151650 , (−6)1430 , (−21)780 }; (ii) {540, 525, 8; 1, 28, 540} with spectrum {5401 , 205400 , (−6)2880 , (−36)2535 }; (iii) {357, 336, 9; 1, 8, 357} with spectrum {3571 , 274914 , (−7)5082 , (−17)5733 }. 4. Classifications of distance-regular graphs with diameter 3 and eigenvalue a2 − c3 In this section, we will classify distance-regular graphs with diameter 3 and eigenvalue a2 − c3 under various conditions. 4.1. k ≤ 2(a1 + 1).

9

In this subsection, we will consider the condition k ≤ 2(a1 + 1), and a classification of distance-regular graphs with diameter 3 and eigenvalue a2 − c3 satisfying k ≤ 2(a1 + 1) is given in the following proposition. Proposition 4.1. Let Γ be a non-bipartite distance-regular graph with valency k, diameter 3 and eigenvalue θ = a2 − c3 . If k ≤ 2(a1 + 1), then Γ is one of the following graphs: (i) The line graph of the Petersen graph with intersection array {4, 2, 1; 1, 1, 4}; (ii) The line graph of the Hoffman-Singleton graph with intersection array {12, 6, 5; 1, 1, 4}; (iii) The regular generalized hexagon of order (4, 1) with intersection array {8, 4, 4; 1, 1, 2}; (iv) The Johnson graph J(7, 3) with intersection array {12, 6, 2; 1, 4, 9}; (v) The halved 7-cube with intersection array {21, 10, 3; 1, 6, 15}. Proof. As Γ is a non-bipartite distance-regular graph with diameter 3 satisfying k ≤ 2(a1 + 1), Proposition 2.6 implies that Γ is either the heptagon, the line graph of a Moore graph of diameter 2, a regular generalized 6-gon of order (s, 1), a Taylor graph, the Johnson graph J(7, 3) or the halved 7-cube. If Γ is the heptagon, then a2 − c3 is equal to −1, but this is not possible by Lemma 3.1. If Γ is the line graph of a Moore graph with valency k  and diameter 2, then Γ has intersection array {2k  − 2, k  − 1, k  − 2; 1, 1, 4} for k  ∈ {3, 7, 57}. By [2, p.149], we find that Γ has an eigenvalue a2 − c3 = k  − 5 when k  ∈ {3, 7}. So, Γ is the line graph of the Petersen graph or the line graph of the Hoffman-Singleton graph. If Γ is a regular generalized 6-gon of order (s, 1), then Γ has intersection array {2s, s, s; 1, 1, 2}. By [2, p.202], we find that s = 4 is the only possible case, and hence Γ is the regular generalized hexagon of order (4, 1). If Γ is a Taylor graph, then by [2, p.431] we know that this is not possible. Both the Johnson graph J(7, 3) and the halved 7-cube have an eigenvalue a2 − c3 = −3 (see [2, p.426-427]). This finishes the proof of the proposition.  With Proposition 4.1, we can strengthen a result of Lemma 3.1 as follows. Lemma 4.2. Let Γ be a non-bipartite distance-regular graph with valency k, diameter 3 and eigenvalue θ = a2 − c3 . Then θ ≤ c23 and a2 ≤ 3c23 . Proof. We need to show that θ = c3 . This will imply θ ≤ c23 , as θ divides c3 (see Lemma 3.1). If θ = c3 , then by Lemma 3.1, a2 = 2c3 , and b1 = c3 + 1. This means that k = c2 + a2 + b2 ≥ 2(c3 + 1) = 2b1 and hence 2(a1 + 1) ≥ k. However, none of the five graphs in Proposition 4.1 have θ = c3 . This shows that θ < c3 . As a2 − c3 = θ ≤ c23 , we find a2 ≤ 3c23 . This finishes the proof.  As a consequence of Proposition 4.1 and Lemma 3.1 we obtain the following proposition. Proposition 4.3. Let Γ be a non-bipartite distance-regular graph with valency k, diameter 3, eigenvalue θ = a2 − c3 . Then the following hold: (i) k ≤ 4c3 ≤ 8a2 ; 10

(ii) If k = 4c3 , then Γ is the generalized hexagon of order (4, 1); c, where c˜ = b1ca2 2 . (iii) k2 < 8˜ Proof. By Lemma 3.1 we obtain b1 ≤ 2c3 and c3 ≤ 2a2 . We first consider the case that k ≥ 2b1 , i.e., k ≤ 2(a1 + 1) holds. As the five graphs listed in Proposition 4.1 all satisfy k ≤ 4c3 with equality if, and only if, Γ is the generalized hexagon of order (4, 1), we find that k ≤ 4c3 . As the generalized hexagon of order (4, 1) satisfies c3 < 2a2 , we have c holds. Now, we assume that k < 2b1 , then we k < 8a2 , and hence k2 = kbc21 = ak˜2c < 8˜ kb1 k˜ c have k < 4c3 ≤ 8a2 and k2 = c2 = a2 < 8˜ c. This finishes the proof.  4.2. Small c3 . In this subsection, we will consider the condition that c3 is small, and a classification of distance-regular graphs with diameter 3 and eigenvalue a2 − c3 satisfying c3 ≤ 9 is given in Theorem 4.7. To give the classification, we need the following results. Lemma 4.4. Let Γ be a non-bipartite distance-regular graph with diameter 3 and eigenvalue θ = a2 − c3 . Then c3 = 1 and, if c3 is a prime, then θ = a2 − c3 = 1 holds. Proof. If c3 is a prime or is equal to one, then we have (a2 − c3 ) ∈ {1, −1, −c3 , c3 }, as / (a2 − c3 )(b1 − c3 ) = c3 (see Lemma 3.1). By Lemma 3.1, we know that, (a2 − c3 ) ∈ {−1, −c3 }. And by Lemma 4.2, we know that θ = a2 − c3 = c3 , i.e., θ = a2 − c3 = 1. This  shows that c3 = 1 and, if c3 is a prime, then θ = a2 − c3 = 1 holds. Lemma 4.5. Let Γ be a non-bipartite distance-regular graph with valency k, diameter 3 and distinct eigenvalues k = θ0 > θ1 > θ2 > θ3 . If θ = a2 − c3 = 1 is an eigenvalue of Γ, then the following hold: (i) θ2 = 1; (ii) θ1 and θ3 are integral; (iii) Γ has intersection array {2c3 + a1 + 1, 2c3 , c3 + a1 − c2 ; 1, c2 , c3 }. Moreover, in this case we obtain the following equation (4.1)

c3 (c2 + 2) = −(θ3 + 1)(θ1 + 1).

Proof. As θ = a2 −c3 = 1, we find θ = θ2 = 1, a2 = c3 +1 and b1 −c3 = c3 , and θ1 and θ3 are integral by Lemma 3.1 and Proposition 3.2. As k = 2c3 + a1 + 1 and a2 = c3 + 1, we obtain b2 = c3 + a1 − c2 , and hence Γ has intersection array {2c3 + a1 + 1, 2c3 , c3 + a1 − c2 ; 1, c2 , c3 }. By Lemma 3.1, we know that θ1 + θ3 = a1 . As θ1 , θ2 and θ3 are eigenvalues of ⎡ ⎤ 0 −1 b1 ⎦, b2 T = ⎣ 1 a1 + 1 − c2 0 c2 a3 − b 2 we find that θ1 θ3 = −a1 − 1 − (c2 + 2)c3 . As a consequence we obtain c3 (c2 + 2) = −(θ3 + 1)(θ1 + 1). 11

 Remark. The intersection array {22 + 1, 2(2 − 1), 2 − 1; 1, 2, 2 − 1} is feasible for any integer  ≥ 2. The distance-regular graph with such an intersection array has a2 −c3 = 1 as an eigenvalue, and a distance-regular graph exists for  = 2, (the Hamming graph H(3, 4)) and  = 4, [13]. For the other values of , it is not known whether a distance-regular graph exists. Proposition 4.6. Let Γ be a distance-regular graph with valency k, diameter 3 and distinct eigenvalues k = θ0 > θ1 > θ2 > θ3 . Let θ = a2 − c3 be an eigenvalue of Γ. If c2 = 1 and c3 is a prime, then Γ is one of the following: (i) Γ is the generalized hexagon of order (4, 1) with intersection array {8, 4, 4; 1, 1, 2}; (ii) Γ is a putative distance-regular graph with intersection array {11, 10, 4; 1, 1, 5}. Proof. As c3 is equal to a prime number p, we find θ2 = 1 = a2 − c3 and θ1 and θ3 are integral by Lemma 4.4 and Lemma 4.5. We note that a regular graph with second largest eigenvalue θ1 ≤ 0 is a complete multipartite graph ([2, Corollary 3.5.4]). That is, we may assume that θ1 > 0. By Eq.(4.1), we see that (θ1 , θ3 ) ∈ {(2, −p − 1), (p − 1, −4), (3p − 1, −2)}. By Lemma 3.1, we know that θ1 + θ3 = a1 . Then (θ1 , θ3 ) = (2, −p − 1) is not possible as 1 − p = θ1 + θ3 = a1 ≥ 0. If (θ1 , θ3 ) = (3p − 1, −2), then we have a1 = θ1 + θ3 = 3p − 3. By Lemma 4.5, we know that k = 2c3 + a1 + 1 = 5p − 2, i.e., k ≤ 2(a1 + 1) holds. This shows that Γ is one of the graphs listed in Proposition 4.1, and only the generalized hexagon of order (4, 1) satisfies that c3 is equal to a prime. If (θ1 , θ3 ) = (p−1, −4), then a1 = θ1 +θ3 = p−5 and k = 2c3 +a1 +1 = 3p−4. We note that if c2 = 1, then for any vertex x of Γ(x) is a disjoint union of cliques of size a1 + 1, i.e., c2 = 1 implies that a1 + 1 divides k ([2, Proposition 4.3.2 and Proposition 4.3.3]). This shows that p − 4 divides 3p − 4. This means that p is equal to 5 and we obtain the feasible intersection array {11, 10, 4; 1, 1, 5}. A distance-regular graph with this intersection array is not known. This finishes the proof.  Now, we give a classification of distance-regular graphs with diameter 3 and eigenvalue a2 − c3 satisfying c3 ≤ 9. Theorem 4.7. Let Γ be a non-bipartite distance-regular graph with diameter 3 and eigenvalue θ = a2 − c3 . If c3 ≤ 9, then Γ is one of the following distance-regular graphs: (i) The generalized hexagon of order (4, 1) with intersection array {8, 4, 4; 1, 1, 2}; (ii) A Hamming or Doob graph with intersection array {9, 6, 3; 1, 2, 3}; (iii) The line graph of the Hoffman-Singleton graph with intersection array {12, 6, 5; 1, 1, 4}; (iv) A putative distance-regular graph with intersection array {11, 10, 4; 1, 1, 5}; (v) A putative distance-regular graph with intersection array {19, 16, 8; 1, 2, 8}; (vi) The Johnson graph J(7, 3) with intersection array {12, 6, 2; 1, 4, 9}; 12

(vii) (viii) (ix) (x)

The Johnson graph J(10, 3) with intersection array {21, 12, 5; 1, 4, 9}; The line graph of the Petersen graph with intersection array {4, 2, 1; 1, 1, 4}; The GQ(2, 4) with a spread removed with intersection array {8, 6, 1; 1, 3, 8}; A distance-regular graph with intersection array {9, 6, 1; 1, 2, 9}.

Proof. Let v be the number of vertices of Γ. We first consider the case where Γ is primitive. Note that if v ≤ 1024, then we could find the above intersection arrays from the tables of [2, p. 425–431]. By Proposition 4.3 and Lemma 3.1, we know that k ≤ 4c3 and b1 ≤ 2c3 . If c2 ≥ 2, then v = 1+k +k2 +k3 = 1+k + kbc21 + kc23b2 ≤ 1+4c3 +4c23 +8c23 , and hence v ≤ 1024 as c3 ≤ 9. So, we may assume that c2 = 1. Then v = 1 + k + k2 + k3 = 1 + k + kbc21 + kc23b2 ≤ 1 + 4c3 + 8c23 + 16c23 holds. As c3 ≤ 6 implies that v ≤ 1024, we assume that c3 ≥ 7. Also, by Proposition 4.6, we know that the case where c2 = 1 and c3 = 7 is not possible. Thus, we only need to consider the following cases: (c2 , c3 ) = (1, 8) and (c2 , c3 ) = (1, 9). By Lemma 3.1, we know that b1 − c3 divides c3 . As b1 ≤ 2c3 , for c3 ∈ {8, 9} we find that 2 b1 = 2c3 or b1 ≤ 12. If b1 ≤ 12, then v = 1 + k + kbc21 + kbc21cb32 ≤ 1 + k + 12k + (12)8 k = 1 + 32k, where the inequality holds as b2 ≤ b1 ≤ 12, c2 = 1 and 8 ≤ c3 ≤ 9. By Lemma 4.2, we know that a2 ≤ 32 c3 ≤ 13.5, and this shows that k = c2 + a2 + b2 ≤ 1 + 13 + 12 = 26. Then we find that v ≤ 1 + 32k ≤ 833 < 1024. So, we may assume that b1 = 2c3 . Note that b1 = 2c3 implies that θ2 = a2 − c3 = 1 (Lemma 3.1). Then by Lemma 4.5, the eigenvalues θ1 and θ3 are integers satisfying 3c3 = (c2 + 2)c3 = −(θ1 + 1)(θ3 + 1) and θ1 + θ3 = a1 ≥ 0. As c2 = 1, a1 + 1 should divide k = 2c3 + a1 + 1, and hence θ1 + θ3 + 1 divides b1 = 2c3 . Then the only solutions are (c3 , θ1 , θ3 ) = (8, 7, −4), and (c3 , θ1 , θ3 ) = (8, 5, −5). This means that k ≤ 20 and v ≤ 1 + 20 + 320 + 400 < 1024. Now we consider the case where Γ is antipodal. Then by Proposition 2.4, there exist integers s, t ≥ 2 satisfying c3 = st. So, we need to consider the following cases: (s, t) ∈ {(2, 2), (2, 3), (3, 2), (2, 4), (4, 2), (3, 3)}. The case (s, t) = (2, 2) belongs to the line graph of the Petersen graph. The case (s, t) ∈ {(2, 3), (3, 2)} is impossible as s + t must divide s(s + 1)t(t + 1). The case (s, t) = (2, 4) is possible, as the pseudoGQ(2, 4) is the unique GQ(2, 4) and this graph has two non-isomorphic spreads. The case (s, t) = (4, 2) belongs to intersection array {8, 4, 1; 1, 1, 8} with 45 vertices, and this intersection array is ruled out by [2, Proposition 4.3.3]. In the last case (s, t) = (3, 3), we find the intersection array {9, 6, 1; 1, 2, 9} for which a distance-regular graph can be constructed by removing a spread of the generalized quadrangle W (3), see [12, p. 46]. This shows the theorem.  4.3. Small a2 . In this subsection, we will consider the condition that a2 is small, and a classification of distance-regular graphs with diameter 3 and eigenvalue a2 − c3 satisfying 1 ≤ a2 ≤ 7 is given in Corollary 4.9. This result follows Lemma 4.8 below. 13

Lemma 4.8. Let Γ be a non-bipartite distance-regular graph with valency k, diameter 3 and eigenvalue θ = a2 − c3 . If c3 = 2a2 and a2 ≤ 8, then Γ is one of the following graphs: (i) The line graph of the Petersen graph with intersection array {4, 2, 1; 1, 1, 4}; (ii) The GQ(2, 4) with a spread removed with intersection array {8, 6, 1; 1, 3, 8}. Proof. Note that, by Lemma 3.1, we know that (a2 − c3 )(b1 − c3 ) = c3 . As c3 = 2a2 , we find that b1 − c3 = −2, i.e., b1 = c3 − 2 = 2(a2 − 1). If k ≤ 2(a1 + 1), then we could check the five graphs in the list of Proposition 4.1, and then we find that Γ is the line graph of the Petersen graph. So, we may assume that k > 2(a1 + 1), i.e., c, k < 2b1 = 4(a2 − 1) < 4a2 holds. As k < 4a2 , we find that k2 = kbc21 < 4ac22b1 = 4˜ 2(a −1)a where c˜ = b1ca2 2 = 2c2 2 ≤ 2(a2 − 1)a2 . Note that b2 ≤ b1 = 2(a2 − 1) and c3 = 2a2 imply k3 = k2 cb23 < k2 < 8a2 (a2 − 1). Let v be the number of vertices of Γ. Then we find v ≤ 4a2 + 2(8a2 (a2 − 1)), and hence v ≤ 1024 holds as a2 ≤ 8. If Γ is primitive, then, by the tables of [2, p. 425–431], we find that such a graph does not exist. So, we may assume that Γ is antipodal. By Proposition 2.4, we have c3 = st and a2 = st − t for some integers s, t ≥ 2. As c3 = 2a2 , we find that s = 2, and hence we have that t ∈ {2, 3, 4} by Lemma 2.3. As s + t divides st(s + 1)(t + 1) (Lemma 2.3), we find that (s, t) ∈ {(2, 2), (2, 4)}. As k > 2(a1 + 1), (s, t) = (2, 2) is not possible. Then, we obtain (ii) in this case.  From the previous result, we obtain a classification of distance-regular graphs with diameter 3 and eigenvalue a2 − c3 satisfying 1 ≤ a2 ≤ 7. Corollary 4.9. Let Γ be a non-bipartite distance-regular graph with diameter 3 and eigenvalue θ = a2 − c3 . If 1 ≤ a2 ≤ 7, then Γ is one of the following distance-regular graphs: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

The generalized hexagon of order (4, 1) with intersection array {8, 4, 4; 1, 1, 2}; A Hamming or Doob graph with intersection array {9, 6, 3; 1, 2, 3}; The line graph of the Hoffman-Singleton graph with intersection array {12, 6, 5; 1, 1, 4}; A putative distance-regular graph with intersection array {11, 10, 4; 1, 1, 5}; The Johnson graph J(7, 3) with intersection array {12, 6, 2; 1, 4, 9}; The line graph of the Petersen graph with intersection array {4, 2, 1; 1, 1, 4}; The GQ(2, 4) with a spread removed with intersection array {8, 6, 1; 1, 3, 8}; A distance-regular graph with intersection array {9, 6, 1; 1, 2, 9}.

Proof. If c3 ≤ 9, then, by checking the list of graphs of Theorem 4.7, we find exactly the above eight graphs. So, we may assume c3 ≥ 10. By Lemma 3.1, we know that a2 ≥ c23 , i.e., a2 ∈ {5, 6, 7}. If c3 = 2a2 , then we are done by Lemma 4.8. So, we may assume 10 ≤ c3 < 2a2 , i.e., a2 ∈ {6, 7}. As a2 − c3 divides c3 we find that 10 ≤ c3 ≤ 32 a2 , and 7α . As α and hence a2 = 7. Set α := b1 − c3 . Then by Lemma 3.1, we find that c3 = 1+α c3 are integers and c3 = 2a2 , we know that c3 ≤ 8, and this contradicts c3 ≥ 10. This completes the proof.  14

4.4. Small c˜. In this subsection, we will consider the condition that c˜ is small, and a classification of distance-regular graphs with diameter 3 and eigenvalue a2 − c3 satisfying c˜ ≤ 26 is given in the following theorem. We recall that a distance-regular graph Γ with diameter 3 has an eigenvalue a2 − c3 if, and only if, the distance-2 graph Γ2 of Γ is a co-edge-regular graph with parameter c˜ = ac22b1 . Theorem 4.10. Let Γ be a non-bipartite distance-regular graph with diameter 3. Assume that the distance-2 graph Γ2 of Γ is a co-edge-regular graph with parameter c˜ = ac22b1 . If c˜ ≤ 26, then Γ is one of the following graphs. (i) The generalized hexagon of order (4, 1) with intersection array {8, 4, 4; 1, 1, 2}; (ii) A Hamming or Doob graph with intersection array {9, 6, 3; 1, 2, 3}; (iii) The Johnson graph J(7, 3) with intersection array {12, 6, 2; 1, 4, 9}; (iv) The line graph of the Petersen graph with intersection array {4, 2, 1; 1, 1, 4}; (v) The GQ(2, 4) with a spread removed with intersection array {8, 6, 1; 1, 3, 8}; (vi) A distance-regular graph with intersection array {9, 6, 1; 1, 2, 9}; (vii) The halved 7-cube with intersection array {21, 10, 3; 1, 6, 15}. Proof. By Proposition 4.3, we know that k ≤ 8a2 , and this implies that k2 = k cb12 ≤ 8 ac22b1 ≤ 8 × 26. If a2 ≤ 7, then we find the first six graphs in the above list by Corollary 4.9. So, we may assume that a2 ≥ 8, and this means that cb12 ≤ 13 . Let v be the number of 4 b1 k2 k3 b2 b2 13 vertices of Γ. Note that c2 = k ≥ k2 = c3 . Set α = c3 ≤ 4 , i.e., k3 = αk2 and k ≤ α1 k2 . 4 Then v = 1 + k + k2 + k3 ≤ 1 + α1 k2 + k2 + αk2 ≤ ( 13 + 1 + 13 )k2 + 1 < 1024, where 4 has the second inequality holds as b1 ≥ c2 and the function f (x) = x1 + x for 1 ≤ x ≤ 13 4 13 maximum value at x = 4 . So, if Γ is primitive, then, by the tables of [2, p. 425–431], we find the seventh graph. Now, we assume that Γ is antipodal. Then, by Proposition 2.4, there exist integers s, t ≥ 2 satisfying c3 = st and c˜ = st(s − 1). If c3 = st ≤ 9, then by Theorem 4.7, we are done. So, we may assume that st ≥ 10, i.e., s ≤ 3 holds as st(s − 1) ≤ 26. If s = 3, then we have 2 ≤ t ≤ 4. If s = 2, then by Lemma 2.3, we find that 2 ≤ t ≤ s2 = 4. Also, Lemma 2.3 says that s + t divides s(s + 1)t(t + 1), and hence we find that t = 3. So we always have st = c3 ≤ 9, and this contradicts st ≥ 10. This finishes the proof.  4.5. Fixed smallest eigenvalue of Γ2 . In this subsection, we will consider the condition that the smallest eigenvalue of Γ2 is fixed, and a classification of distance-regular graphs with diameter 3 and eigenvalue a2 − c3 such that its distance-2 graphs have the smallest eigenvalue −2 or −3 is given in Proposition 4.12. We recall that a distance-regular graph Γ with diameter 3 has an eigenvalue a2 − c3 if, and only if, the distance-2 graph Γ2 of Γ is a connected strongly 15

regular graph. We first introduce some properties of distance-regular graphs with diameter 3 and eigenvalue a2 − c3 when the distance-2 graphs have the smallest eigenvalue m. Theorem 4.11. Let Γ be a non-bipartite distance-regular graph with diameter 3, eigenvalue a2 − c3 and v vertices. Assume that the distance-2 graph Γ2 of Γ is a connected ˜ a strongly regular graph with parameters (˜ v = v, k, ˜, c˜) and integral eigenvalues k˜ > σ > τ = −m. Then the following hold: (i) k˜ < 8˜ c; (ii) c˜ ≤ (2m − 3)m3 ; (iii) If c˜ = m2 , then k˜ ≤ 8m2 and v ≤ 64m(m + 1); (iv) If c˜ = m(m − 1), then k˜ < 8m(m − 1) and v ≤ 64(m − 1)2 . Proof. (i) As k˜ = k2 , this follows from Proposition 4.3. (ii) This follows immediately from Theorem 2.1. (iii) Let c˜ = m2 . Then by Theorem 2.2, Γ2 is a Steiner graph. This shows that there w(w−1) , and k˜ = m w−m ([11, p.395-396]). exists a positive integer w such that v = v˜ = m(m−1) m−1 ˜ we find w ≤ m(8m − 7), and hence v ≤ 64m(m + 1). As 8m2 > k, (iv) If c˜ = m(m − 1), then by Theorem 2.2, Γ2 is a Latin Square graph. This shows that v = v˜ = (s + 1)2 and k˜ = ms for some positive integer s ([11, p.395-396]). As  k˜ < 8m(m − 1), we see s < 8(m − 1) and hence v ≤ 64(m − 1)2 . Now, we give a classification of distance-regular graphs with diameter 3 and eigenvalue a2 − c3 such that its distance-2 graphs have the smallest eigenvalue −2 or −3 . Proposition 4.12. Let Γ be a non-bipartite distance-regular graph with diameter 3, eigenvalue a2 − c3 and v vertices. Assume that the distance-2 graph Γ2 of Γ is a connected ˜ a strongly regular graph with parameters (˜ v = v, k, ˜, c˜) and smallest eigenvalue −m. (i) If m = 2, then Γ is the line graph of the Petersen graph with intersection array {4, 2, 1; 1, 1, 4} or the GQ(2, 4) with a spread removed with intersection array {8, 6, 1; 1, 3, 8}. (ii) If m = 3, then Γ is one of the following distance-regular graphs: (a) The Johnson graph J(7, 3) with intersection array {12, 6, 2; 1, 4, 9}; (b) A distance-regular graph with intersection array {9, 6, 1; 1, 2, 9}; (c) A distance-regular graph with intersection array {15, 12, 1; 1, 4, 15}; (d) The GQ(3, 9) with a spread removed with intersection array {27, 24, 1; 1, 8, 27}. Proof. If c˜ ≤ 26, then by Theorem 4.10, we obtain the two graphs in (i) for m = 2 and the first two graphs in (ii) for m = 3. We note that by [2, Theorem 3.12.4], we know that the case where m = 2 is finished. So, we may assume that c˜ ≥ 27 and m = 3. By Theorem 2.1, we find that c˜ ≤ 81, i.e., 27 ≤ c˜ ≤ 81. For m = 3 and 27 ≤ c˜ ≤ 81, by using computer ˜ a we find feasible parameters (˜ v , k, ˜, c˜) for connected strongly regular graphs with smallest eigenvalue −3 having at least 1025 vertices. By Theorem 2.2, second largest eigenvalues 16

σ of such strongly regular graphs satisfy σ ≤ 12 m(m − 1)(˜ c + 1) − 1. And we obtained that the only possible strongly regular graph must have parameters (1911, 270, 105, 27). But this case satisfies k˜ > 8˜ c, and this is not possible by Theorem 4.11. So, we may assume that Γ has at most 1024. If Γ is primitive, then from the tables of [2, p. 425–431], we find, besides the Johnson graph J(7, 3), a putative distance-regular graph with intersection array {19, 12, 5; 1, 4, 15}. It was shown by Coolsaet and Jurisi´c [5] that such a distance-regular graph does not exist, see also [14]. Now, we assume that Γ is antipodal. Then by Proposition 2.4, there exist integers s, t ≥ 2 satisfying k = st with s = 3, 2 ≤ t ≤ 9 and s+t divides s(s+1)t(t+1) (Lemma 2.3). So t = 3, 5, 6 or 9. Haemers [8] showed that there does not exist a pseudoGQ(3, 6). So the case t = 6 is not possible. This shows the proposition.  4.6. Smallest eigenvalue > −3. In this subsection, we will consider the condition that the smallest eigenvalue is larger than −3, and a classification of distance-regular graphs with diameter 3 and eigenvalue a2 −c3 satisfying θ3 > −3 is given in Theorem 4.14. We first look at the following theorem by Bang and Koolen which gives a classification of distance-regular graphs with valency at least 3, diameter 3 and the smallest eigenvalue θ3 > −3. Theorem 4.13. ([1, Theorem 1.3]) Let Γ be a distance-regular graph with valency k ≥ 3 and diameter 3. If the smallest eigenvalue θ3 satisfies θ3 > −3, then one of the following holds: (i) Γ is the icosahedron; (ii) Γ is the Klein graph with intersection array {7, 4, 1; 1, 2, 7}; (iii) Γ is the P G(2, 8) with intersection array {8, 6, 1; 1, 1, 8}; (iv) Γ is a generalized hexagon of order (s, 1), where s ≥ 2; (v) Γ is the line graph of a Moore graph. Now, we give a classification of distance-regular graphs with diameter 3 and eigenvalue a2 − c3 satisfying θ3 > −3. Theorem 4.14. Let Γ be a non-bipartite distance-regular graph with diameter 3 and eigenvalue θ = a2 − c3 . If the smallest eigenvalue θ3 satisfies θ3 > −3, then one of the following holds: (i) Γ is the generalized hexagon of order (4, 1); (ii) Γ is the line graph of the Petersen graph; (iii) Γ is the line graph of the Hoffman-Singleton graph. Proof. Note that by Lemma 3.1, we know that the distance-2 graph Γ2 of Γ is strongly regular. We may assume that the valency of Γ is at least 3 as the distance-2 graph of a 17

heptagon is again a heptagon. As θ3 > −3, we know that Γ is one of the graphs listed in Theorem 4.13. As the distance-2 graphs of the icosahedron, the Klein graph with intersection array {7, 4, 1; 1, 2, 7} and the P G(2, 8) with intersection array {8, 6, 1; 1, 1, 8} are not strongly regular, Γ is none of them. Assume that Γ is a generalized hexagon of order (s, 1) with s ≥ 2. Then Γ has intersection array {2s, s, s; 1, 1, 2} and its distance-2 graph is strongly regular if, and only if, 2 = c3 = (a2 − c3 )(b1 − c3 ) = (s − 3)(s − 2) holds, i.e., s ∈ {1, 4}. As Γ is non-bipartite, s is not equal to 1, and hence Γ is the generalized hexagon of order (4, 1). Now, we assume that Γ is the line graph of a Moore graph with valency k (and diameter 2). Then Γ has intersection array {2k − 2, k − 1, k − 2; 1, 1, 4} and its distance-2 graph is strongly regular if, and only if, 4 = c3 = (a2 − c3 )(b1 − c3 ) = (k − 5)2 holds, i.e., k ∈ {3, 7}. So, we find that Γ is either the line graph of the Petersen graph or the line graph of the Hoffman-Singleton graph. This finishes the proof.  4.7. Remarks on diameter 4. We finish this paper with some remarks on the distance-2 graph of a distance-regular graph with diameter 4. Remark 4.15. (i) The distance-2 graph of a non-bipartite antipodal distance-regular graph with diameter 4 has exactly 4 distinct eigenvalues (see, [2, Corollary 4.2.5]); (ii) For diameter 4, there are only two feasible intersection arrays in the table of [2] such that their distance-2 graphs are strongly regular graphs. The two graphs are the Hamming graph H(4, 3) with intersection array {8, 6, 4, 2; 1, 2, 3, 4}, and a putative distance-regular graph with intersection array {39, 32, 20, 2; 1, 4, 16, 30}. It was shown by Lambeck [10] that such a putative distance-regular graph does not exist.

Acknowledgments Q. Iqbal and M.U. Rehman are supported by Chinese Scholarship Council at USTC, Hefei, China. J.H. Koolen has been partially supported by the National Natural Science Foundation of China (Grants No. 11471009 and No. 11671376) and by the Anhui Initiative in Quantum Information Technologies (Grant No. AHY150200). This work was (partially) done while J. Park was working at Wonkwang University. Also, J. Park is supported by Basic Research Program through the National Research Foundation of Korea funded by Ministry of Education (NRF-2017R1D1A1B03032016). References [1] S. Bang, J.H. Koolen, Distance-regular graphs of diameter 3 having eigenvalue −1, Linear Algebra Appl., 531:38–53, 2017. 18

[2] A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance-regular graphs, Springer-Verlag, Berlin, 1989. [3] A.E. Brouwer, S.M. Cioab˘a, F. Ihringer and M. McGinnis, The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters, J. Combin. Theory Ser. B, 133:88–121, 2018. [4] A.E. Brouwer, Distance-regular graphs of diameter 3 and strongly regular graphs, Discrete Math., 49:101–103, 1984. [5] K. Coolsaet, A. Jurisi´c, Using equality in the Krein conditions to prove the nonexistence of certin distance-regular graphs, J. Combin. Theory Ser. A, 115:1086–1095, 2008. [6] C. Dalf´ o, M.A. Fiol, J.H. Koolen, The spectral excess theorem for graphs with few eigenvalues whose distance-2 or distance-1-or-2 graph is strongly regular, arXiv:1608.00373, DOI:10.1080/03081087.2018.1491944. [7] C.D. Godsil, G.F. Royle, Algebraic graph theory, GTM 207, Springer-Verlag, New York, 2001. [8] W.H. Haemers, There exists no (76,21,2,7) strongly regular graph. Finite geometry and combinatorics (Deinze, 1992), 175–176, London Math. Soc. Lecture Note Ser., 191, Cambridge Univ. Press, Cambridge, 1993. [9] J.H. Koolen, J. Park, Distance-regular graphs with a1 or c2 at least half the valency, J. Combin. Theory Ser. A, 119:546–555, 2012. [10] E. Lambeck, Some elementary inequalities for distance-regular graphs, European J. Combin., 14:53, 1993. [11] A. Neumaier, Strongly regular graphs with smallest eigenvalue −m, Arch. Math. (Basel), 33:392–400, 1979/1980. [12] S.E. Payne, J.A. Thas, Finite generalized quadrangles, second ed., in: EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Z¨ urich, 2009, p.xii+287. [13] M. Shi, D. Krotov and P. Sol´e, A new distance-regular graph of diameter 3 on 1024 vertices, Des. Codes Cryptogr., 87:2091–2101, 2019. [14] E.R van Dam, J.H. Koolen, H. Tanaka, Distance-regular graphs, Electron. J. Combin., DS22, 2016.

19