arc-transitive distance-regular covers of cliques with λ = µ

ISSN 0081-5438, Proceedings of the Steklov Institute of Mathematics, 2014, Vol. 284, Suppl. 1, pp. S124–S134.c© Pleiades Publishing, Ltd., 2014.Original Russian Text c© A.A. Makhnev, D.V. Paduchikh, L.Yu. Tsiovkina, 2013,published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2013, Vol. 19, No. 2.

Arc-Transitive Distance-Regular Coversof Cliques with λ = μ

A. A. Makhnev1,2, D. V. Paduchikh1, and L. Yu. Tsiovkina1

Received December 14, 2012

Abstract—We study antipodal distance-regular graphs of diameter 3 such that their auto-morphism group acts transitively on the set of pairs (a, b), where {a, b} is an edge of the graph.Since the automorphism group of such graphs acts 2-transitively on the set of antipodal classes,the classification of 2-transitive permutation groups can be used. We classify arc-transitivedistance-regular graphs of diameter 3 in which any two vertices at distance at most two haveexactly μ common neighbors.Keywords: arc-transitive graphs, antipodal distance-regular graphs, automorphism groups.

DOI: 10.1134/S0081543814020114

INTRODUCTION

We consider undirected graphs without loops and multiple edges. For a vertex a of a graph Γ,we denote by Γi(a) the i-neighborhood of the vertex a, i.e., the subgraph induced by Γ on the setof all vertices at distance i from a. Denote [a] = Γ1(a) and a⊥ = {a} ∪ [a].

The degree of a vertex is the number of vertices in its neighborhood. A graph Γ is called a regulargraph of degree k if the degree of any vertex from Γ equals k. A graph Γ is called an edge-regulargraph with parameters (v, k, λ) if it contains v vertices, is regular of degree k, and any of its edgeslies in λ triangles. A graph Γ is called an amply regular graph with parameters (v, k, λ, μ) if it isedge-regular with the corresponding parameters and [a]∩ [b] contains μ vertices for any two verticesa and b at distance 2 in Γ. An amply regular graph of diameter 2 is called a strongly regular graph.

If vertices u and w are at distance i in Γ, then we denote by bi(u,w) (by ci(u,w)) the numberof vertices in the intersection of Γi+1(u) (Γi−1(u)) and [w]. A graph Γ of diameter d is called adistance-regular graph with intersection array {b0, b1, . . . , bd−1; c1, . . . , cd} if the values bi(u,w) andci(u,w) are independent of the choice of vertices u and w at distance i in Γ for any i = 0, . . . , d. Letai = k − bi − ci. Note that, for a distance-regular graph, b0 is the degree of the graph and c1 = 1.

A graph Γ of diameter d is called distance-transitive if, for any i ∈ {0, . . . , d} and any two pairsof vertices (u,w) and (y, z) with d(u,w) = d(y, z) = i, there exists an automorphism g of Γ suchthat (ug, wg) = (y, z). For a subset X of automorphisms of a graph Γ, we denote by Fix(X) theset of all vertices of Γ fixed by any automorphism from X. Further, we denote by pl

ij(x, y) the

1Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi 16,Yekaterinburg, 620990 Russiaemails: [email protected], [email protected], [email protected]

2Institute of Radioelectronics and Informational Technologies, Ural Federal University, ul. Mira 19, Yekaterinburg,620002 Russia

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ARC-TRANSITIVE COVERS OF CLIQUES S125

number of vertices in the subgraph Γi(x) ∩ Γj(y) for vertices x and y at distance l in the graph Γ.In a distance-regular graph, the numbers pl

ij(x, y) are independent of the choice of vertices x and y.These numbers are denoted by pl

ij and are called the intersection numbers of Γ.Denote by GX the group of permutations of G on a set X. If Y ⊆ X, then GY (G{Y }) is the

pointwise (global) stabilizer of Y in G.

In [1], a description of antipodal distance-transitive graphs of diameter 3 was obtained. Theproblem of describing arc-transitive antipodal distance-regular graphs of diameter 3 is more general.

A graph is called arc-transitive if its automorphism group acts transitively on the set of its arcs(ordered edges).

Proposition 1 [1, Lemma 2.7]. Suppose that a noninvariant subgroup H of a group G andan element g ∈ G are given. Denote by Γ = Γ(G,H,HgH) the graph with the vertex set V (G,H) ={Hx | x ∈ G} and edges {Hx,Hy} such that xy−1 ∈ HgH. Then,

(1) if G acts faithfully on V (G,H), g2 ∈ H and G = 〈H, g〉, then Γ is a connected graph,G ≤ Aut(Γ), and G acts faithfully and transitively on vertices and arcs of the graph Γ;

(2) if G acts transitively on arcs of a connected graph Δ, H is the stabilizer of a vertex e ∈ Δ,g is a 2-element, and the vertices e and eg are adjacent in Δ, then Δ ∼= Γ(G,H,HgH), g2 ∈ H,and G = 〈H, g〉;

(3) if H < M < G, g2 ∈ H, G = 〈H, g〉, and the conditions:(i) G = M ∪ MgH,(ii) Hg ∩ M ≤ H,(iii) H ∩ Hg acts transitively on Γ3(H) = {Hm | m ∈ M − H}

hold, then Γ = Γ(G,H,HgH) is a cover of the complete graph on V (G,M) and G is a distance-transitive automorphism group of the graph Γ.

An antipodal distance-regular graph Γ of diameter 3 has (see [2]) intersection array {k, μ(r−1), 1;1, μ, k}; v = r(k +1) vertices; and spectrum k1, nf , (−1)k, (−m)h, where n and −m are roots of theequation x2 + (μ − λ)x − k = 0, f = m(r − 1)(k + 1)/(n + m), and h = n(r − 1)(k + 1)/(n + m).

Let δ = k − 1− rμ = a1 − c2. In [3], it is proved that, for fixed r and δ, there exist only finitelymany admissible arrays, except for the cases δ ∈ {−2, 0, 2}. It is also established in [3] that, ifδ ∈ {−2, 2}, then k + 1 is a square.

If μ = λ, then δ = 0 and Γ has intersection array {rμ+1, μ(r− 1), 1; 1, μ, rμ+1}; v = r(rμ+2)

vertices; and spectrum k1,√

kf, (−1)k, (−

√k)f , where f =

(r

2

)μ + r − 1 = (v − k − 1)/2. In what

follows, the number μ(r − 1) is even.In this paper, we propose a program for investigating arc-transitive antipodal distance-regular

graphs of diameter 3 based on the classification of 2-transitive permutation groups. We implementa part of this program for graphs with λ = μ. The results of this paper were announced in [4].

Remark. Suppose that Γ is an antipodal distance-regular graph of diameter 3, G = Aut(Γ),Σ is the set of all antipodal classes of Γ, and F ∈ Σ. Then, the following statements are equivalent:

(1) Γ is an arc-transitive graph, and G acts 2-transitively on Σ;(2) Γ is a vertex-transitive graph and, for a vertex a ∈ F , the group Ga acts transitively

on Σ − {F}.Let Γ be an arc-transitive graph, and let a ∈ F . Then, Ga acts transitively on [a]. Therefore, Ga

acts transitively on Σ − {F}, and G acts 2-transitively on Σ.

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Conversely, suppose that Γ is a vertex-transitive graph; G acts transitively on Σ; and, for avertex a ∈ F , the group Ga acts transitively on Σ − {F}. Then, for any K,L ∈ Σ − {F}, thereexists g ∈ Ga such that L = Kg. Hence, for a unique vertex b ∈ [a] ∩ K, we obtain bg ∈ [a] ∩ L.Therefore, Ga acts transitively on [a].

Recall that |G{F} : Ga| = r. If r is relatively prime to k (for example, if k = rμ + 1), then, forthe arc-transitivity of Γ, it is sufficient that Γ be vertex-transitive and G act 2-transitively on Σ.

Proposition 2 [1, Theorem 2.9]. Suppose that GX is a 2-transitive permutation group ofdegree n, a ∈ X, H = Ga, and T is the socle of the group G. Then, either

(1) G is an almost simple group and one of the following possibilities holds for (T, n):

(i) the alternating groups (An, n), n ≥ 5;(ii) the linear groups (Lm(q), (qm − 1)/(q − 1)), m ≥ 2, (m, q) /∈ {(2, 2), (2, 3)};(iii) the symplectic groups (Sp2m(2), 22m−1 ± 2m−1), m ≥ 3;(iv) the unitary groups (U3(q), q3 + 1), q ≥ 3;(v) the Ree groups (2G2(q), q3 + 1), q = 32a+1 ≥ 27;(vi) the Suzuki groups (Sz(q), q2 + 1), q = 22a+1 ≥ 8;(vii) the Mathieu groups (Mn, n), n ∈ {11, 12, 22, 23, 24};(viii) the sporadic groups (L2(11), 11), (M11, 12), (A7, 15), (Aut(L2(8)), 28), (HiS, 176), (Co3, 276)

or

(2) G = TH, T is an elementary abelian group of order n = pm, and one of the followingpossibilities holds:

(i) the linear groups with m = cd, d ≥ 2, and SLd(pc) � H ≤ ΓLd(pc);(ii) the symplectic groups with m = ct, t is even, t ≥ 4, and Spt(pc) � H ≤ Zpc−1 ◦ ΓSpd(pc);(iii) the G2-type groups with m = 6c, p = 2, and G2(2c)′ � H ≤ Z2c−1 ◦ Aut(G2(2c));(iv) the scalar groups with H ≤ ΓL1(pm);(v) the exceptional groups with pm ∈ {92, 112, 192, 292, 592} and SL2(5) � H, orpm = 24 and A6 or A7 � H, orpm = 36 and SL2(13) � H;(vi) the extraspecial groups with pm ∈ {52, 72, 112, 232} and SL2(3) � H or pm = 34, R =

D8 ◦ Q8 � H, H/R ≤ S5, and 5 divides |H|.Let us list new infinite series of antipodal distance-regular graphs of diameter 3 constructed

based on Propositions 1 and 2:1. The Suzuki series. The graph Suz(q) has intersection array {q2, q2 − q − 2, 1; 1, q + 1, q2},

where q = 22e+1 ≥ 8, r = q − 1, G = Aut(Suz(q)) = Aut(Sz(q)), and Ga is an extension of a Sylow2-subgroup P of order q2 by the cyclic group of order 2e + 1.

There is a simple representation of the graph Suz(q). The vertices of the graph are involutionsof the group Sz(q), and two involutions are adjacent if the order of their product is 5. Therefore,the neighborhood of a vertex in the graph Suz(q) has a partition in 4-cliques.

If the graph Suz(q) exists, then, for any divisor s of the number q − 1 such that 1 < s ≤ q − 1,there exists the graph Suz(q, s) with intersection array {q2, (s − 1)(q2 − 1)/s, 1; 1, (q2 − 1)/s, q2};moreover, G = Aut(Suz(q, s)) = Aut(Sz(q)), and Ga is an extension of a Sylow 2-subgroup P oforder q2 by an abelian group of order (q − 1)(2e + 1)/s.

2. The Ree series. The graph Ree(q) has intersection array {q3, q3 − 2q2 − 2q − 3, 1; 1, 2(q2 +q + 1), q3}, where q = 32e+1 > 3, r = (q − 1)/2, G = Aut(Ree(q)) = Aut(2G2(q)), and Ga is



an extension of a Sylow 3-subgroup P (the socle of the group) of order q3 by the cyclic group oforder 4e + 2.

If the graph Ree(q) exists, then, for any divisor s of the number (q − 1)/2 such that 1 <

s ≤ (q − 1)/2, there exists a graph Ree(q, s) with intersection array {q3, (s − 1)(q3 − 1)/s, 1; 1,(q3 − 1)/s, q3}, where q = 32e+1 > 3, G = Aut(Ree(q, s)) = Aut(2G2(q)), and Ga is an extension ofa Sylow 3-subgroup P (the socle of the group) of order q3 by a group of order (q − 1)(2e + 1)/s.

3. The unitary series. The graph U3(q) has intersection array {q3, (r − 1)(q3 − 1)/r, 1; 1,(q3 − 1)/r, q3}, where q = pe > 3, r is the 2′-part of the number (q − 1), G = Aut(U3(q)) =Aut(U3(q)), and Ga is an extension of a Sylow p-subgroup P (the socle of the group) of order q3

by an abelian group of order (q2 − 1)e/r.If the graph U3(q) exists, then, for any nontrivial divisor s of the 2′-part of the number (q − 1),

there exists a graph U3(q, s) with intersection array {q3, (s− 1)(q3 − 1)/s, 1; 1, (q3 − 1)/s, q3}, whereq = pe > 3, G = Aut(U3(q, s)) = Aut(U3(q)), and Ga is an extension of a Sylow p-subgroup P (thesocle of the group) of order q3 by an abelian group of order (q2 − 1)e/s.

Theorem 1. Suppose that Γ is an arc-transitive distance-regular graph with intersection array{k, (r − 1)μ, 1; 1, μ, k} and G = Aut(Γ). Then,

(1) if Γ is a bipartite graph, then it is obtained by deleting a maximum matching from Kk+1,k+1

and G ≤ 2 × Sk+1;

(2) if r = k, then either k = 2 and Γ is a hexagon or k = 6, Γ is the second neighborhood of avertex in the Hoffman–Singleton graph, and G ≤ S7;

(3) if r = 2 < k, then, up to the passage to the graph Γ2, one of the following possibilities holds :

(i) k + 1 = 22m−1 ± 2m−1, G ≤ 2 × Sp2m(2), m ≥ 3, and μ = 22m−2;(ii) k = q3, G ≤ 2 × PΓU3(q), q > 3 or G ≤ 2 × Aut(2G2(q)), q = 32e+1, e ≥ 1, andμ = (q + 1)(q2 − 1)/(2, q − 1);

(iii) k = q, G ≤ 2 × PΣL2(q), q ≡ 1 (mod 4), and μ = (q − 1)/2;(iv) k = 175, μ = 72, and G = 2 × HiS or k = 275, μ = 112, and G = 2 × Co3;(v) k = 22t − 1, G = 2 × ASp2t(2), and μ = 22t−1.

In case (3), the graph Γ2 is also an arc-transitive distance-regular graph with μ′ = k − μ − 1.In case (ii), a misprint was made in the main theorem from [1] for the Ree group 2G2(q): there

should be n = 36e+3 + 1 instead of n = 32e+1 + 1.

Theorem 2. Suppose that Γ is an arc-transitive distance-regular graph with intersection array{rμ + 1, (r − 1)μ, 1; 1, μ, rμ + 1} and G = Aut(Γ). If r > 2, then

(1) k = q is a power of a prime, r divides q − 1, Γ has intersection array {q, (r − 1)(q −1)/r, 1; 1, (q − 1)/r, q}, and L2(q) � G;

(2) Γ ∈ {Suz(q, s),Ree(q, s),U3(q, s)}.

1. AUXILIARY RESULTS

This section contains results used in the proof of the theorem.

Lemma 1. Let OK be the ring of algebraic integers of a field K. If an integer d is not amultiple of a squared prime and K = Q(d1/2) is the corresponding quadratic field, then the integerbasis of the ring OK is (1, (1 + d1/2)/2) if d ≡ 1 (mod 4) and (1, d1/2) if d ≡ 2, 3 (mod 4).

Proof. This statement is Exercise 4 from [5, Ch. 2].


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Lemma 2. Suppose that Γ is a distance-regular graph with intersection array {k, μ(r − 1), 1;1, μ, k}, G = Aut(Γ), g ∈ G, and Ω = Fix(g) is a nonempty graph. Then,

(1) if Ω intersects antipodal classes K and L, then |K ∩ Ω| = |L ∩ Ω|;(2) if Ω is a nonempty graph and g fixes an antipodal class K, then K intersects Ω;(3) if g is an element of prime order p and p > μ, then Ω is a clique, or Ω is contained in an

antipodal class, or Ω is a distance-regular graph with intersection array {t−1, μ(s−1), 1; 1, μ, t−1};(4) if g is an element of prime order p, λ = μ, and p does not divide r − 1, then any vertex

from Ω is adjacent to some vertex from Γ − Ω.

Proof. Suppose that Ω intersects antipodal classes K and L. Then, a vertex from L ∩ Ω liesin the neighborhood of a unique vertex from K ∩ Ω; hence, |K ∩ Ω| ≤ |L ∩ Ω|. Symmetrically,|L ∩ Ω| ≤ |K ∩ Ω|.

Suppose that Ω is a nonempty graph and g fixes an antipodal class K. Choose an antipodalclass L containing a vertex a from Ω. Then, a vertex from [a] ∩ K lies in Ω.

Suppose that Γ has t antipodal classes intersecting Ω at s vertices. If s = 1, then Ω is a clique.If t = 1, then Ω is contained in an antipodal class. If s, t > 1, then Ω is an antipodal s-cover of at-clique. Since p > μ, we have μΩ = μ; hence, Ω is a distance-regular graph with intersection array{t − 1, μ(s − 1), 1; 1, μ, t − 1}.

Suppose that g is an element of prime order p, a⊥ ⊆ Ω, and p does not divide r − 1. Then,Γ3(a) contains a vertex b from Ω. Note that [b] ⊂ Ω since, otherwise, for a vertex x ∈ [b] − Ω, thesubgraph [x] ∩ [xg] contains the vertex b and μ vertices from [a], a contradiction. Now, for a vertexy ∈ Γ2(a) ∩ Γ2(b) − Ω, we obtain |[y] ∩ [yg]| ≥ 2μ, a contradiction. Thus, Γ2(a) ∩ Γ2(b) ⊂ Ω andΓ = Ω, a contradiction. The lemma is proved.

The proof of Theorem 2 is based on Higman’s method of dealing with automorphisms of distance-regular graphs, which is presented in Chapter 3 of Cameron’s monograph [6]. We consider thegraph Γ as a symmetric scheme of relations (X,R) with d classes, where X is the vertex set ofthe graph; R0 is the equality relation on X; and, for i ≥ 1, the class Ri consists of pairs (u,w)such that d(u,w) = i. For u ∈ Γ, we define ki = |Γi(u)|, v = |X|. The class Ri corresponds tothe graph Γi on the vertex set X such that vertices u and w are adjacent if (u,w) ∈ Ri. Supposethat Ai is the adjacency matrix of the graph Γi for i > 0 and A0 = I is the identity matrix. Then,AiAj =

∑pl

ijAk, where plij are the intersection numbers of the graph Γ.

Assume that Pi is the matrix whose (j, l)-entries are the numbers plij. Then, the eigenvalues

p1(0), . . . , p1(d) of the matrix P1 are the eigenvalues of the graph Γ with multiplicities m0 =1, . . . ,md. The matrices P and Q whose (j, l)-entries are the numbers pj(i) and qj(i) = mjpi(j)/ni,respectively, are called the first and the second eigenvalue matrices of the scheme; they are connectedby the equality PQ = QP = |X|I.

Suppose that uj and wj are the left and the right eigenvectors of the matrix P1 that correspondto the eigenvalue p1(j) and have the first coordinate 1. Then, wj are the columns of P and mjuj

are the rows of Q.The permutation representation of the group G = Aut(Γ) on vertices of the graph Γ gives

in a standard way a matrix representation ψ of the group G in GL(n, C). The space Cn is theorthogonal direct sum of proper G-invariant subspaces W0, . . . ,Wd of the adjacency matrix A = A1

of the graph Γ. For any g ∈ G, the matrix ψ(g) is permutable with A; hence, the subspace Wi

is ψ(G)-invariant. Let χi be the character of the representation ψWi . Then (see [6, Sect. 3.7]), for



g ∈ G, we obtain

χi(g) = n−1d∑

j=0

Qijαj(g),

where αj(g) is the number of points x from X such that (x, xg) ∈ Rj.

Lemma 3. Assume that Γ is a distance-regular graph that has intersection array {rμ + 1,

μ(r − 1), 1; 1, μ, rμ + 1}; v = r(rμ + 2) vertices; and spectrum k1,√

kf, (−1)k, (−

√k)f , where k =

rμ + 1 and f =(

r

2

)μ + r − 1. Let G = Aut(Γ). If g ∈ G, χ1 is the character of the projection of

the representation ψ to a subspace of dimension f (corresponding to the eigenvalue θ1), and χ2 isthe character of the projection of the representation ψ to a subspace of dimension k, then

χ1(g) = ((r − 1)α0(g) + ((r − 1)α1(g) − α2(g))/√

k − α3(g))/(2r),

χ2(g) = (α0(g) + α3(g))/r − 1.

If |g| = p is prime, then χ2(g) − k is a multiple of p.

Proof. We have

P1 =

⎛⎜⎜⎜⎝

0 1 0 0k μ μ 00 k − μ − 1 k − μ − 1 k

0 0 1 0

⎞⎟⎟⎟⎠ .

Consider, for example, p1(1) =√

k. Then,

P1 −√

kI =

⎛⎜⎜⎜⎝

−√

k 1 0 0k μ −

√k μ 0

0 (r − 1)μ (r − 1)μ −√

k k

0 0 1 −√

k

⎞⎟⎟⎟⎠ .

If (1, x2, x3, x4) is a row vector from the kernel of the matrix P1 −√

kI, then x2 = 1/√

k, x3 =−μ/(

√k(k − μ − 1), and x4 = −μ/(k − μ − 1). Hence,

Q =

⎛⎜⎜⎜⎝

1 1 1 1f f/

√k −(rμ/2 + 1)/

√k −(rμ/2 + 1)

k −1 −1 k

f −f/√

k (rμ/2 + 1)/√

k −(rμ/2 + 1)

⎞⎟⎟⎟⎠ .

Therefore, χ1(g) = ((r − 1)α0(g) + (r − 1)α1(g)/√

k − α2(g)/√

k − α3(g))/(2r).Similarly, χ2(g) = (kα0(g) − α1(g) − α2(g) + kα3(g))/v. Substituting α1(g) + α2(g) = v−

α0(g) − α3(g), we obtain χ2(g) = (α0(g) + α3(g))/r − 1.The remaining statements of the lemma follow from [7, Lemma 1].

Lemma 4 [1, Theorem 2.5]. Suppose that Γ is a distance-regular nonbipartite graph withintersection array {k, μ(r−1), 1; 1, μ, k}, K is an abelian subgroup from Aut(Γ) that is transitive onany antipodal class, and p is a prime divisor of r. Then, p divides k +1 and, in the case k = rμ+1,the number r is a power of 2.


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Proof of Theorem 1. If Γ is a bipartite graph, then, by [1, Corollary 2.3], Γ is obtained bydeleting a maximum matching from Kk+1,k+1, and G ≤ 2 × Sk+1.

If r = k, then k ∈ {2, 6, 56} and Γ is the second neighborhood of a vertex u in the Mooregraph Δ. If k = 2, then Γ is a hexagon; if k = 6, then Γ is the second neighborhood of a vertex inthe Hoffman–Singleton graph and G ≤ S7. If k = 56, then G is a subgroup from the automorphismgroup of the graph Δ acting transitively on Δ − u⊥, which contradicts the theorem from [8].

If r = 2, then the graph Γ is distance-transitive, and we can apply the main theorem from [1].Theorem 1 is proved.

2. AUTOMORPHISMS OF A GRAPH WITH INTERSECTION ARRAY

{rμ + 1, (r − 1)μ, 1; 1, μ, rμ + 1}

Suppose that Γ is an arc-transitive distance-regular graph with intersection array {rμ + 1,(r − 1)μ, 1; 1, μ, rμ + 1} for r > 2, G = Aut(Γ), g ∈ G, and Ω = Fix(g). Note that the graph Γcontains rμ + 2 antipodal classes, each containing r vertices. Suppose that Σ is the set of antipodalclasses of the graph Γ, K is the kernel of the action of G on Σ, F ∈ Σ, a ∈ F , and C is the kernelof the action of G{F} on F .

If μ = 1, then the existence of the graph Γ is equivalent to the existence of a Desarguesianprojective plane of order k (see [2, Sect. 1.17]). Therefore, k = 2e and L2(k) � G.

In this section, we assume that μ > 1 and k = l2d, where d is not a multiple of a squared prime.

Lemma 5. The following statements hold:(1) if d �= 1 and g is an automorphism of Γ of order 2, 3, or 5, then either p = d = 5 or

(r − 1)α1(g) = α2(g) and χ1(g) = ((r − 1)α0(g) − α3(g))/(2r);

(2) if Ω is an empty graph and |g| = p is prime, then either(i) p does not divide r, α3(g) = 0, α1(g) + α2(g) = v, χ1(g) = (rα1(g) − v)

√k/(2kr), and

α1(g) − 1 is a multiple of ld; if d �= 1 and (α1(g) − v/r)/(ld) is odd, then d equals 1 modulo 4 or(ii) p divides r, α3(g) = tr, and χ1(g) = ((r−1)α1(g)−α2(g))

√k/(2rk)− t/2; if d �= 1 and t

is odd, then d equals 1 modulo 4 and the number ((r − 1)α1(g) − α2(g))/(lrd) is odd; if α3(g) = v,then χ1(g) = −(rμ + 2)/2, k is odd, and μ is even;

(3) if Ω contains t antipodal classes, then α3(g) = 0, α1(g) + α2(g) = r(k + 1 − t), andχ1(g) = (r − 1)t/2 + (α1(g)(r − 1) − (k + 1 − t))

√k/(2k).

Proof. If g is an element of a group, then the value of the character χi(g) is the sum of n rootsof unity of degree |g|, where n is the degree of the representation of the projection of ψ to Wi.

Note that roots of unity of degree 2 and 3 have rational real parts. If |g| = 2, 3, then thevalue of the character is real; hence, it is integer. In the case |g| = 5, we can use the formulascos 2π/5 = (

√5 − 1)/4 and cos 4π/5 = −(

√5 + 1)/4.

Thus, if g is an automorphism of the graph Γ of order 2, 3, or 5, then it follows from Lemmas 1and 3 that either (r − 1)α1(g) = α2(g) and χ1(g) = ((r − 1)α0(g) − α3(g))/(2r) or p = d = 5,K = Q(51/2), and the integer basis of the ring OK equals (1, (1 +51/2)/2). Statement (1) is proved.

Suppose that Ω is an empty graph and |g| = p is prime. If p does not divide r, then α3(g) = 0,α1(g) + α2(g) = v, χ1(g) = (rα1(g) − v)

√k/(2kr), and α1(g) − 1 is a multiple of ld. In view of

Lemma 1, if the number (α1(g) − v/r)/(ld) is odd, then d is not equal to 1 modulo 4.Let p divide r. Then, α3(g) = tr and χ1(g) = ((r − 1)α1(g)−α2(g))

√k/(2rk)− t/2. If t is odd,

then, in view of Lemma 1, we have (r − 1)α1(g) �= α2(g), the number d equals 1 modulo 4, andχ1(g) = a + b(1 + d1/2)/2; hence, a + b/2 = −t/2 and b = ((r − 1)α1(g) − α2(g))/(lrd) is odd.



If α3(g) = v, then χ1(g) = −(rμ + 2)/2, k is odd, and λ = μ is even. Statement (2) is proved.Let Ω contain t antipodal classes. It follows from Lemma 2 that α3(g) = 0, |g| divides k + 1− t,

α1(g) + α2(g) = r(k + 1 − t), and χ1(g) = (r − 1)t/2 + (α1(g)(r − 1) − (k + 1 − t))√

k/(2k).

Lemma 6. G/K is an almost simple group.

Proof. Consider the affine case. Let T be the socle of the group G = G/K. If |T | = 2n, then k isodd, μ is even, and rμ = 2n−2; hence, r is odd. Let N be a Sylow 2-subgroup from T . By Frattini’sargument, we can assume that NG(N) contains Ga. Further, there are r N -orbits of length 2n. If theorbit aN contains a vertex from [a], then it follows from the transitivity of Ga on [a] that [a] ⊂ aN ,a contradiction. Let g be an involution from N . Then, α2(g) = v and χ1(g) = −(k + 1)

√k/(2k), a

contradiction with Lemma 1.Suppose that |T | = pn and p is odd. Then, rμ = pn − 2; hence, r is relatively prime to p. Let N

be a Sylow p-subgroup from T . By Frattini’s argument, we can assume that NG(N) contains Ga.Let g be an element of order p from N . As above, α2(g) = v and χ1(g) = −(k + 1)

√k/(2k), a

contradiction with Lemma 1.

Lemma 7. If K = 1, then one of the following statements holds:(1) k = q is a power of a prime, Γ has intersection array {rμ + 1, (r − 1)μ, 1; 1, μ, rμ + 1},

and L2(q) � G;(2) Γ ∈ {Suz(q, r),Ree(q, r),U3(q, r)}.Proof. Let K = 1. In view of Lemma 6, the socle T of the group G is a simple group. If T

acts intransitively on the vertex set of the graph Γ, then the number of T -orbits divides r. Ifthe orbit aT contains a vertex adjacent to a, then it follows from the transitivity of Ga on [a]that [a] ⊂ aT , a contradiction. Hence, aT is a coclique of an order that is a multiple of s(k + 1).On the other hand, in view of Hoffman’s bound, the order of a coclique in Γ is not greater thanr(k + 1)

√k/(k +

√k) = r

√k − r(k −

√k)/(k +

√k). Hence, s(k + 1) < r

√k.

If T = An for n = rμ + 2, then G{F} ∈ {An−1,Sn−1} and |G{F} : Ga| = r, a contradiction.If T = Lm(q), n = (qm − 1)/(q − 1), q = pe, and m ≥ 2, then r divides (qm − 1)/(q − 1)− 2 and

T{F} is the semidirect product of an elementary abelian group N of order qm−1 and H, where H isan extension of SLm−1(q) by the cyclic group of order (q − 1)/(m, q − 1). Hence, N is contained inthe kernel C of the permutation group GF

{F}. In the case m = 2, we have k = q and L2(q) � G.In the case m ≥ 3, the stabilizer of a nonzero vector e1 acts transitively on the set of vectors

not lying in 〈e1〉; hence, the diameter of the graph Γ equals 2, a contradiction.If T = Sp2m(2), n = 22m−1±2m−1, and m ≥ 3, then r is an odd divisor of 22m−1±2m−1−2 and

|T{F} : O±2m(2)| = 2, a contradiction with the fact that the number 22m−2 ± 2m−2 − 1 is relatively

prime to 22i − 1 for i ∈ {1, . . . ,m}.If T = U3(q) and rμ = q3 − 1, then T{F} is an extension of a group P of order q3 by the cyclic

group of order (q2−1)/(3, q+1); |G{F}| divides |T{F}|2e, where q = pe; G{F} contains a subgroup Ga

of index r; and r divides (q − 1)(2e, q2 + q + 1). Since k + 1 ≥ r√

k, then T acts transitively on thevertex set of the graph Γ, and we can assume that T = G. Then, r divides q − 1. If r is even, thenthere exists a quotient of the graph Γ with antipodal classes of order 2. This contradicts Theorem 1;hence, r is odd. Further, Ga acts transitively on Σ − {F} and Γ = U3(q, r).

If T = 2G2(q)′, rμ = q3 − 1, and q = 32e+1 ≥ 3, then T{F} is an extension of a group of order q3

by the cyclic group of order (q − 1), |G{F}| divides |T{F}|(2e + 1), G{F} contains a subgroup Ga of

index r, and r divides (q − 1)(2e + 1, q2 + q + 1). Since k + 1 ≥ r√

k, then T acts transitively on the


S132 MAKHNEV et al.

vertex set of the graph Γ, and we can assume that T = G. Then, r divides (q − 1); in particular,e > 0. Further, Ga acts transitively on Σ − {F}, and Γ = Ree(q, r).

If T = Sz(q), rμ = q2 − 1, and q = 22e+1 ≥ 8, then T{F} is an extension of a group of order q2

by the cyclic group of order (q − 1), |G{F}| divides |T{F}|(2e + 1), G{F} contains a subgroup Ga of

index r, and r divides (q−1)(2e+1, q+1). Since (r, q−1)(k+1) ≥ r√

k, then T acts transitively onthe vertex set of the graph Γ, and we can assume that T = G. Then, r divides (q − 1). Further, Ga

acts transitively on Σ − {F}, and Γ = Suz(q, r).If T is a Mathieu group Mn, where rμ + 2 = n ∈ {11, 12, 22, 23, 24}, then G{F} ∈ {Mn−1,

Aut(Mn−1)} and |G{F} : Ga| = r. In the case n = 11, we have r = 3 = μ and M′10

∼= L2(9), acontradiction. In the case n = 12, we have r = 5 and μ = 2, a contradiction. In the case n = 22,we have (μ, r) ∈ {(2, 10), (4, 5)}. In view of [9, Proposition 1], the first case is impossible. In thesecond case, we have a contradiction with the fact that M′

21∼= L3(4). In the case n = 23, we have

rμ = 21, a contradiction with the fact that M22 does not contain subgroups of index not greaterthan 7. Finally, in the case n = 24, we have r = 11 and μ = 2, a contradiction as above.

In the sporadic case, if T = L2(11), then G{F} = A5 and r = 3 = μ, a contradiction. If T = M11

and n = 12, then G{F} = L2(11), r = 5, and μ = 2, a contradiction. If T = A7 and n = 15, thenrμ = 13, a contradiction. If T = L2(8) and n = 28, then r = 13 and μ = 2, a contradiction. IfT = HiS and n = 176, then r = 87 and μ = 2, a contradiction. If T = Co3 and n = 276, thenr = 137 and μ = 2, a contradiction.

Lemma 8. If K �= 1, then |K| = 2 and T = K × L2(q).

Proof. Let K �= 1. By Lemma 5, k is odd and μ is even. It follows from the transitivity of theaction of G{F} on F that |K| divides r. Consider the graph Γ whose vertices are K-orbits and edgesare defined as follows: two vertices aK and bK are adjacent if and only if some vertex from aK isadjacent to a vertex from bK . If |K| < r, then Γ is an arc-transitive antipodal graph of diameter 3with intersection array {r′μ + 1, (r′ − 1)μ, 1; 1, μ, r′μ + 1}, where r′ = r/|K| and μ = |K|μ.

Assume that G = KY , where Y is a component of the group G that acts transitively on thevertex set of the graph Γ. Let h ∈ K and a ∈ F . Then, ah = ay for an appropriate y ∈ Y ; hence,yh−1 ∈ Ga = Ya and h ∈ Y . Consequently, K ⊂ Y , and we can assume that G = Y . In the case|K| = r, by Lemma 4, the number r is a power of 2.

If T = An and n = rμ + 2 is even, then G{F}/K ∈ {An−1; Sn−1}; |G{F}/K : GaK/K| = r/|K|;and either r = |K| or r/|K| = 2, G{F}/K = Sn−1, and GaK/K = An−1. Since |K| < n − 1, T

contains a component Y that is a covering group for An, and Y acts transitively on the vertex setof the graph Γ. Now, we can assume that G = Y . If n = 6, then r = μ = 2, a contradiction. Hence,n ≥ 8, |K| = 2, and r = 4. Therefore, Γ is an arc-transitive antipodal graph of diameter 3 withintersection array {2μ + 1, μ, 1; 1, μ, 2μ + 1}, a contradiction with Theorem 1.

Suppose that T = Lm(q), n = (qm − 1)/(q − 1), m ≥ 2, q = pe, p is an odd prime, m is even, r

divides (qm−1)/(q−1)−2, and T{F}/K is the semidirect product of an elementary abelian group N

of order qm−1 and an extension H of SLm−1(q) by the cyclic group of order (q − 1)/(m, q − 1). Inview of [10], it follows from the action of T on K that T contains a component Y that is a coveringgroup for Lm(q).

If m ≥ 4, then Y acts transitively on the vertex set of the graph Γ, and Ya contains the semidirectproduct of an elementary abelian group E of order qm−1 and a subgroup L isomorphic to SLm−1(q).In view of Lemma 7, we can assume that Y is not a simple group, G = Y , and GF

{F} is an extensionof the central subgroup of order s dividing (m, q − 1) by the cyclic group of order (q − 1)/s. Since



((qm − 1)/(q − 1)− 2, q − 1) = (m− 2, q − 1), we have |K| = 2. Considering the graph Γ, we obtaina contradiction with Lemma 7.

Thus, m = 2 and r divides q−1. Recall that the order of the Schur multiplier of the group L2(q)equals 2 if q is odd and equals 6 if q = 9. Hence, |K| = 2 and the graph Γ has intersection array{q, (r′ − 1)2μ, 1; 1, 2μ, q}, where r′ = r/2. By [2, Remark (iii) to Proposition 12.5.3], we concludethat either q and r are odd and G = L2(q).Z2 or q is odd, r is even, and G = Z2 × L2(q).

If T = Sp2m(2), n = 22m−1 ± 2m−1, and m ≥ 3, then r is an odd divisor of 22m−1 ± 2m−1 − 2and |T{F}/K : O±

2m(2)| = 2. In view of [10], it follows from the action of T on K that T containsa component Y that is a covering group for Sp2m(2). Repeating the argument from the precedingparagraph, we find that r = 4, a contradiction.

If T = U3(q) and rμ = q3 − 1, then q = pe, p is an odd prime, T{F}/K is an extension of agroup P of order q3 by the cyclic group of order (q2−1)/(3, q +1), |G{F}| divides |T{F}|2e, G{F}/K

contains a subgroup GaK/K of index r, and r divides (q − 1)(2e, q2 + q + 1). If q = 5, then, inview of [10], the minimal degree of a faithful permutation representation of the group U3(q) is 50and rμ = 124. Therefore, in view of [10], it follows from the action of T on K that T contains acomponent Y that is a covering group for U3(q). Recall that the order of the Schur multiplier of thegroup U3(q) divides 3; a contradiction with the fact that 3 = (3, q +1) divides (q−1)(2e, q2 +q +1).

If T = 2G2(q)′, rμ = q3 − 1, and q = 32e+1 ≥ 3, then T{F}/K is an extension of a group oforder q3 by the cyclic group of order (q − 1), |G{F}| divides |T{F}|(2e + 1), G{F}/K contains asubgroup GaK/K of index r, and r is an odd divisor of (q− 1)(2e+1, q2 + q +1). In the case q = 3,we have r = 13 and μ = 2, a contradiction. Hence, e > 0 and T contains a component Y of type2G2(q), a contradiction with the fact that the Schur multiplier of the group 2G2(q) is trivial.

Suppose that T is a Mathieu group Mn, where rμ + 2 = n ∈ {12, 22, 24}. Then, G{F}/K ∈{Mn−1,Aut(Mn−1)} and |G{F}/K : GaK/K| = r/|K|. In the case n = 12, we have r = 5 = |K|and μ = 2, a contradiction with Lemma 4. In the case n = 22, we have (μ, r) ∈ {(2, 10), (4, 5)}.In view of [9, Proposition 1], the first case is impossible. In the second case, M′

21∼= L3(4) and, in

view of Lemma 4, we have r = 10, μ = 2, and |K| = 5, a contradiction with the fact that order ofthe Schur multiplier of the group M22 is not a multiple of 5. Finally, in the case n = 24, we haver = 11 = |K| and μ = 2, a contradiction with Lemma 4.

In the sporadic case, if T/K ∼= M11 and n = 12, then r = 5 = |K| and μ = 2. If T/K ∼= L2(8)and n = 28, then r = 13 = |K| and μ = 2. If T/K = HiS and n = 176, then r = 87 = |K|and μ = 2. If T/K = Co3 and n = 276, then r = 137 = |K| and μ = 2. In any case, we have acontradiction with Lemma 4.

The lemma and Theorem 2 are proved.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research (project no. 12-01-00012), by a joint grant of the Russian Foundation for Basic Research and the National NaturalScience Foundation of China (project no. 12-01-91155), by a program of the Division of MathematicalSciences of the Russian Academy of Sciences (project no. 12-T-1-1003), and by programs of jointresearch of the Ural Branch of the Russian Academy of Sciences with the Siberian Branch of theRussian Academy of Sciences (project no. 12-S-1-1018) and the Belorussian National Academyof Sciences (project no. 12-S-1-1009). The first author was also supported by the Program forState Support of Leading Universities of the Russian Federation (agreement no. 02.A03.21.0006 ofAugust 27, 2013).


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Translated by E. Vasil’eva


arc-transitive distance-regular covers of cliques with λ = µ

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