J Algebr CombDOI 10.1007/s10801-014-0566-x

Two new infinite families of arc-transitive antipodaldistance-regular graphs of diameter three withλ = μ related to groups Sz(q) and 2G2(q)

L. Yu. Tsiovkina

Received: 1 August 2013 / Accepted: 15 October 2014© Springer Science+Business Media New York 2014

Abstract In this paper, we construct two new infinite families of arc-transitive distan-ce-regular graphs, related to Suzuki groups Sz(q) and Ree groups 2G2(q), whereq > 3. They are antipodal r -covers of complete graphs on q2 + 1 or q3 + 1 vertices,respectively, with λ = μ and r > 1 being an arbitrary odd divisor of q − 1. We alsofind that the graph on the set of involutions of Sz(q) with q > 3, whose edges are thepairs of involutions {u, v} such that |uv| = 5, is distance-regular.

Keywords Antipodal graph · Arc-transitive graph · Automorphism group ·Distance-regular graph · r -Fold covering graph

Mathematics Subject Classification 05E18 · 05E30

1 Introduction

Let Γ be an antipodal distance-regular graph of diameter three. Then Γ is an antipodalr -cover of the complete graph Kk+1, Γ has intersection array {k, μ(r −1), 1; 1, μ, k},and any two adjacent vertices of Γ have exactly λ = k − 1 − μ(r − 1) commonneighbours, where parameter k is the valency of Γ and μ denotes the number ofcommon neighbours for any two vertices of Γ at distance 2 (for notation, definition,and further theory of distance-regular graphs, see [2,12]). For more background onantipodal distance-regular covers of complete graphs, we refer the reader to [4]. Thissubclass of imprimitive distance-regular graphs has been studied by many researchersfor over the last twenty years. A very informative overview of the state of the art on such

L. Yu. Tsiovkina (B)Department of Algebra and Topology, Krasovsky Institute of Mathematicsand Mechanics of UB RAS, 16, S.Kovalevskaya street, Yekaterinburg 620990, Russiae-mail: l.tsiovkina@gmail.com

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covers can be found in the recent paper [6]. Even though the problem of classificationof antipodal distance-regular covers of complete graphs seems to be hopeless, a studyof some of their subclasses with prescribed automorphism groups or combinatorialproperties may provide further understanding of their structure.

Antipodal distance-transitive covers of complete graphs were classified in [5]. Amore general problem is to describe all arc-transitive antipodal distance-regular coversof complete graphs (recall that a graph is called arc-transitive if its automorphismgroup acts transitively on ordered pairs of adjacent vertices).

An arc-transitive antipodal distance-regular graph Γ of diameter three possessesan automorphism group which acts 2-transitively on the set of fibres. It follows fromthe fact that any pair of distinct fibres (F1, E1) in Γ can be mapped to any otherpair of distinct fibres (F2, E2) by an automorphism which maps an arc (x1, y1) to anarc (x2, y2) with xi ∈ Fi , yi ∈ Ei , i = 1, 2, see [5, Lemma 2.6]. The assumptionof distance-transitivity involves another 2-transitive action (which, in general, is notnecessary in the arc-transitive case), namely, 2-transitive action of the global stabilizerof an arbitrary fibre F in the full automorphism group on F . These observationstogether with the classification of finite 2-transitive permutation groups (for example,see [5, Theorem 2.9]) lead to some strong restrictions for the automorphism groupsof such graphs.

In [8], a description of arc-transitive antipodal distance-regular graphs of diameterthree with λ = μ is given. In particular, in [8] the authors found some necessaryconditions characterizing three new potential families of such graphs correspondingto groups U3(q), Sz(q) and 2G2(q). The existence of arc-transitive candidates forthese families is provided by the coset graph construction. However, it was not knownwhether they are distance-regular or not (except for some distance-regular examplesfor small values of q which were constructed by the authors of [8] in GAP).

In this paper, we show that in several cases these arc-transitive graphs are indeeddistance-regular, namely, we find two new infinite families of distance-regular graphsrelated to Suzuki groups Sz(q) and Ree groups 2G2(q), where q = 22n+1 > 2 orq = 32n+1 > 3, respectively. A technique of proving their distance-regularity seemsto be new and uses a canonical form of elements in groups Sz(q) and 2G2(q), whichin turn determines the structure of matchings in these graphs. In Sect. 2, we havecollected some auxiliary results, and Sect. 3 contains the main results and their proofs.

2 Preliminaries

In this section, we provide some necessary fundamental results on arc-transitivegraphs, coverings and groups Sz(q) and 2G2(q). Our notation and terminology fromgroup theory are standard and can be found in [1,3].

Proposition 1 [5, Lemma 2.7] Suppose that a non-normal subgroup H of a groupG and an element g ∈ G are given. Let Γ (G, H, HgH) denote the graph withvertex set V (G, H) = {H x | x ∈ G} whose edges are the pairs {H x, H y} such thatxy−1 ∈ HgH.

(1) Assume that G acts faithfully on V (G, H), g2 ∈ H and G = 〈H, g〉. ThenΓ (G, H, HgH) is a simple, undirected, and connected graph which admits the

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group G acting (by right multiplication) faithfully and transitively both on verticesand on arcs.

(2) Suppose G acts arc-transitively on a connected graph X, H is the stabilizer of avertex x in G, and g ∈ G is a 2-element interchanging x with some vertex adjacentto x in X. Then X � Γ (G, H, HgH), and also g2 ∈ H and G = 〈H, g〉.

Recall the notion of a covering graph according to [4]. Let Γ be a simple, undirectedgraph and suppose there is a partition Σ of its vertices into cells satisfying the followingconditions: (1) each cell is an independent set; (2) between any two cells either thereare no edges, or there is a matching. Let Γ/Σ be the graph with the cells of Σ asvertices and with two vertices adjacent if and only if there is a matching betweenthem. Then Γ is said to be a covering graph of Γ/Σ . The elements of Σ are calledfibres and if all of them have the same size r , then Γ is called r -fold covering graphof Γ/Σ .

Further in the paper for a fixed graph Γ and for any vertex a, we will denote by [a]the set of neighbours of a in Γ .

Proposition 2 Let H and S be non-normal subgroups of a group G, let M = NG(S)

and let g be an element of G such that S ≤ H < M < G, g2 ∈ H, G = 〈H, g〉, G =M ∪ MgS, H g ∩ M ≤ H, Mg ∩S = 1 and Γ = Γ (G, H, HgH). Suppose in additionthat |G : M | = k + 1, |M : H | = r, gcd(r, k) = 1, |S| = k = |H : H g ∩ H |. Then Γ

is an arc-transitive r-fold covering graph of Γ/Σ � Kk+1 with Σ = {{Hm(gs)e| m ∈M}|s ∈ S, e ∈ {0, 1}}.Proof Proposition 2 is a modified form of Lemma 2.7 of [5]. For completeness, weprovide its proof. Let us consider the following four H -invariant sets Γ0 = {H}, Γ1 ={Hgh|h ∈ H}, Γ2 = {H x |x /∈ gH ∪ M} and Γ3 = {Hm|m ∈ M − H}. Clearly, onlyΓ2 or Γ3 might not be H -orbits. Since g2 ∈ H, H H g = Hg2g−1 Hg = HgHg. Wehave H ≤ M , and then the inclusion H g ∩ M ≤ H can be multiplied on the left andright by H . Thus

H(H g ∩ M) = (H(g−1 Hg)) ∩ M = HgHg ∩ M ⊆ H, (1)

and((H(g−1 Hg)) ∩ M)H = HgHgH ∩ M ⊆ H. (2)

Further, the right H -cosets at distance at most 2 from H in Γ are precisely thosein HgHgH ∪ HgH . Suppose that m ∈ M − H and vertices Hm and H are atdistance less than 3. Then m ∈ HgHgH ∪ HgH . If m ∈ HgHgH , then by (2)m ∈ H , a contradiction. If m ∈ HgH , then g ∈ M , but G = 〈H, g〉, a contradiction.Consequently, the set Γ3 = {Hm | m ∈ M − H} consists of vertices at distance atleast 3 from H in Γ .

As the inclusion (1) can be multiplied on the right by g, we see that (HgHg ∩M)g = HgH ∩ Mg ⊆ Hg. This implies that there is only edge in Γ between H and{Hmg| m ∈ M}, that is the edge {H, Hg}. Let m1, m2 ∈ M − H . If Hm1 ∈ [Hm2]then m1m−1

2 ∈ HgH and g ∈ M , but G = 〈H, g〉, a contradiction. Suppose that thevertices Hm1, Hm2 are at distance 2 in Γ . Then m1m−1

2 ∈ HgHgH . By (2), we havethat m1m−1

2 ∈ H , which implies Hm1 = Hm2, a contradiction.

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The valency of Γ is equal to the number of right H -cosets in the double coset HgH ,that is to |H : H g ∩ H | = |S| = k. Thus every vertex from Γ3 is adjacent precisely tok vertices from Γ2. As H g ∩S = 1, S acts regularly on [H ], and G acts arc-transitivelyon Γ . This implies that there exists a constant λ such that any two adjacent verticeshave exactly λ common neighbours. The stabilizer of the arc (H, Hg) in G is preciselyH g ∩ H .

As given, the group M acts transitively on {Mx | x ∈ G−M}, |G : M | = k+1, |M :H | = r and gcd(r, k) = 1. We have |M : Mg ∩ M | = k and r = |Mg ∩ M : H g ∩ H |.

This implies that Mg ∩ M-orbit on the arcs of Γ , containing the arc (H, Hg), hassize r . Hence there is a matching between {Hm| m ∈ M} and {Hmg| m ∈ M} in Γ

which is Mg ∩ M-invariant, and Mg ∩ M acts transitively on {Hm| m ∈ M}.Since M = NG(S), it follows that every element of S fixes Γ0 ∪ Γ3 pointwise.Let Σ be the partition of vertices of Γ into k + 1 cells of type {Hm(gs)e| m ∈ M},

where s ∈ S and e ∈ {0, 1}. It follows that Σ forms an imprimitivity system of G onthe vertices of Γ , such that each of its blocks corresponds to a right M-coset of Gwith appropriate representative (gs)e.

Since Γ0 ∪ Γ3 consists of vertices at pairwise distance at least 3 in Γ , for eachelement s ∈ S, the set (Γ0 ∪ Γ3)gs also consists of vertices at pairwise distance atleast 3 in Γ , and there is a matching between {Hm| m ∈ M} and {Hmgs| m ∈ M}in Γ . Thus each cell of Σ is an independent set of size r . Note also, that S acts regularlyon the neighbourhood of any vertex from Γ0 ∪ Γ3. Therefore, G acts 2-transitively onΣ , and there is a matching between any two cells of Σ . Hence Γ is an r -fold coveringgraph of Kk+1. This completes the proof. �Proposition 3 [9,10] Let G = Sz(q), where q > 2 is an odd power of 2, S ∈Syl2(G), M = NG(S), and let g be an involution contained in G−S. Then G = 〈S, g〉and the following statements hold.

(1) M = S : K , where K = M ∩ Mg, 1 = S ∩ Mg, K � Zq−1 and |S| = q2.(2) G = M ∪ MgS and each element in G − M can be written uniquely in the form

xgy, where x ∈ M and y ∈ S.(3) K 〈g〉 � D2(q−1).(4) S contains exactly q − 1 involutions. All involutions of S generate Z(S) and

|Z(S)| = q.(5) There is a single class of conjugate involutions in G and CG(a) = S for any

involution a ∈ S.

Proposition 4 [7,11] Let G = 2G2(q), where q > 3 is an odd power of 3, S ∈Syl3(G), M = NG(S), and let g be an involution contained in G−M. Then G = 〈S, g〉and the following statements hold.

(1) M = S : K , where K = M ∩ Mg, 1 = S ∩ Mg, K � Zq−1 and |S| = q3.(2) G = M ∪ MgS and each element in G − M can be written uniquely in the form

xgy, where x ∈ M and y ∈ S.(3) K = A0 × 〈h〉, where h is the unique involution of K , (q − 1)/2 is odd, and

NG(A0) � D2(q−1).(4) There is a single class of conjugate involutions in G.

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(5) CS(h) = CS1(h) is an elementary abelian group of order q and CS(h)∩ Z(S) = 1with S1 = S(1).

(6) CS(x) = 1 for each x ∈ A0.

3 Main results

Theorem 1 Let G ∈ {Sz(q), 2G2(q)}, where q = p2n+1 > 3, S ∈ Sylp(G) withp = 2 if G = Sz(q), and p = 3 if G = 2G2(q). Let g be an involution of G notcontained in NG(S), let 〈h〉 be a subgroup of NG(S) ∩ NG(S)g of odd index r > 1and H = S〈h〉. Then Γ (G, H, HgH) is an arc-transitive antipodal distance-regulargraph with intersection array {q2, (q2−1)(r−1)/r, 1; 1, (q2−1)/r, q2} if G = Sz(q),or {q3, (q3 − 1)(r − 1)/r, 1; 1, (q3 − 1)/r, q3} if G = 2G2(q), and Γ (G, H, HgH)

does not depend (up to isomorphism) on the choice of involution g ∈ G − M.

Proof We put M = NG(S), and let r be the maximal odd divisor of |M ∩ Mg|. Wehave M ∩ Mg = 〈h〉 × A, where |A| = r and |h| = (q − 1)/r . Let H = S〈h〉,Γ = Γ (G, H, HgH), A = 〈 f 〉, and L = 〈 f 〉〈g〉. Then L � D2r .

Further, let Σ = {{Hm(gs)e| m ∈ M}| s ∈ S, e ∈ {0, 1}}. By Propositions 3, 4and 2, it follows that Γ is an arc-transitive r -fold covering graph of Γ/Σ � K|S|+1.Now, we show that there exists a constant μ such that any two vertices from differentfibres of Γ have exactly μ common neighbours. In order to do this, we will considerthe action of L on the set of vertices of Γ .

By F(H x), we will denote the fibre of Γ which contains the vertex H x . We haveF(H) = {H f i }r

i=1 and F(Hg) = {Hg f i }ri=1.

For convenience, we will also use the following notation for the vertices of fibresF(H) and F(Hg):

a1 = H, a2 = H f, ..., ar = H f r−1

and

b1 = Hg, b2 = Hg f, ..., br = Hg f r−1.

It is clear that ai ∈ [bi ] for all i ∈ {1, .., r}. For a fibre F ∈ Σ , an element y ∈ G,and a vertex H x = a ∈ F , we set a y = H xy and Fy = F(a y). If Δ is a subset ofvertices of Γ , then by Δg we denote the set of images of vertices of Δ under the actionof g.

Recall that the stabilizer of any three points in G in its doubly transitive represen-tation is trivial if G = Sz(q), see [9] or [10], or has a unique nontrivial element fixingmore than two points (namely, it fixes exactly q + 1 points) if G = 2G2(q), see [11]or [7]. Since any two involutions are conjugate in G, it follows that the element gfixes (pointwise) exactly w fibres in Γ with w = 1 if G = Sz(q) and w = q + 1 ifG = 2G2(q). �

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Lemma 1 The number of 〈 f 〉-orbits on Σ −{F(H), F(Hg)} is equal to (|S|− 1)/r .There is only one L-orbit of size 2 on Σ , namely {F(H), F(Hg)}. There are exactlyw L-orbits of size r on Σ , and the size of any other L-orbit on Σ is 2r .

Proof This follows from Propositions 3 and 4. �Lemma 2 Suppose Lr is an L-orbit on Σ of size r and define Δ to be the set of allvertices which lie in the union of fibres from Lr . Then, for all i, j ∈ {1, .., r}, eachvertex b j is adjacent to only one vertex in [ai ] ∩ Δ.

Proof Let us consider the action of L on Δ. The element g fixes only one fibre in Lr ,which is denoted further by F1. Every 〈 f 〉-orbit on Δ contains only one vertex fromany fibre in Δ, and the neighbourhood of any vertex from F(H) contains only onevertex from any 〈 f 〉-orbit on Δ. The element g fixes each 〈 f 〉-orbit on Δ setwise byfixing just one vertex (which lies in F1) in it.

Let F1 = F(Hgs) = {Hg f j s}rj=1 for some s ∈ S. Then Hg f j sg = Hg f j s for

all j ∈ {1, .., r}. We have

(a j+1g, b j+1g

) =(

H f j g, Hg f j g)

=(

Hg f − j , H f − j)

= (br− j+1, ar− j+1

)

for all 1 ≤ j < r and(a1g, b1g

) = (b1, a1

). Hence

Hg f j s = Hg f j sg ∈[

H f j]

∩[

H f j g]

= [a j+1

] ∩ [br− j+1

]

for all 1 ≤ j < r . As a j+1 = a1 f j and br− j+1 = b1 f − j , we have

Hg f j s f − j ∈ [a j+1 f − j ] ∩ [br− j+1 f − j ] = [a1] ∩ [b1 f −2 j ]

for all 1 ≤ j < r . Assume that b1 f −2 j1 = b1 f −2 j2 for some j1, j2 ∈ {1, .., r} suchthat j1 < j2. Then 2( j2 − j1) = lr for some integer l, hence l = 0, a contradiction.

Hence for all j ∈ {1, .., r}, each vertex b j is adjacent to only one vertex in [a1]∩Δ.By a similar argument as above, we obtain that for all i, j ∈ {1, .., r}, each vertex b j

is adjacent to only one vertex in [ai ] ∩ Δ. �Lemma 3 Suppose L2r is an L-orbit on Σ of size 2r and define Δ to be the set of allvertices which lie in the union of fibres from the same 〈 f 〉-orbit on L2r . Then, for alli, j ∈ {1, .., r}, each vertex b j is adjacent to only one vertex in [ai ] ∩ Δ and to onlyone vertex in [ai ] ∩ Δg.

Proof Let D = Δ ∪ Δg. Now consider the action of L on D. We choose two verticesHgs, Hgs1 ∈ D, where elements s, s1 ∈ S are such that fibres F(Hgs), F(Hgs1)

belong to distinct 〈 f 〉-orbits on Σ . There is an involution g f α , where α ∈ {1, .., r},in L such that F(Hgsg f α) = F(Hgs1). Clearly F(Hg f α) = F(Hg).

Consider the structure of the matching between fibres F(Hg) and F(Hgs1). Wehave Hg f i sg f α ∈ [H f i g f α] for all i ∈ {1, .., r}, {Hg f j s1}r

j=1 = {Hg f j sg f α}rj=1.

Furthermore, Hgs1 = Hg f βsg f α for some β ∈ {1, .., r}. The element gsg ∈ G − M

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can be written uniquely in the form mgs0 with m ∈ M, s0 ∈ S. Assume that m =s2 f δhγ , where s2 ∈ S, δ ∈ {1, .., r}, γ ∈ {1, .., (q − 1)/r}. Then

Hg f βsg f α = H f −βgsg f α = H f −βmgs0 f α = H f −βs2 f δhγ gs0 f α

= H f −β f δg f α s0 = H f δ−β−αgs0 = Hgs1

for some s0 ∈ S. It follows that s0 = s1 and f δ−β−α ∈ H . Hence f αs1 = s0 f α andr divides δ − β − α.

Further,

Hg f t sg f α = H f −t gsg f α = H f −t mgs0 f α = H f −t s2 f δhγ g f αs1

= H f δ−α−t gs1

for any t ∈ {1, .., r}. Then

Hg f i s1 = Hg f j sg f α

for some i, j ∈ {1, .., r} if and only if

H f δ−α− j gs1 = H f −i gs1

if and only if r divides δ − α − j + i − (δ − β − α) = β + i − j. Hence

Hg f i s1(= Hg f β+i sg f α) ∈ [H f β+i g f α] ∩ [H f i ] = [b1 f α−β−i ] ∩ [a1 f i ]

and

Hg f β+i s ∈ [H f β+i ] ∩ [H f i g f α] = [a1 f β+i ] ∩ [b1 f α−i ]

for all i ∈ {1, .., r}. Then for all i ∈ {1, .., r}, the following inclusions hold

Hg f i s1 f −i ∈ [b1 f α−β−2i ] ∩ [a1]

and

Hg f β+i s f −(β+i) ∈ [a1] ∩ [b1 f α−β−2i ].

Suppose that b1 f α−β−2i1 = b1 f α−β−2i2 for some i1, i2 ∈ {1, .., r} such that i1 < i2.Then α − β − 2i1 = α − β − 2i2 + lr for an integer l, hence 2(i2 − i1) = lr andl = 0, a contradiction. Consequently, for all j ∈ {1, .., r} every vertex b j is adjacentto only one vertex in [a1] ∩ Δ and to only one vertex in [a1] ∩ Δg. By an argumentas above, we have that for all i, j ∈ {1, .., r} every vertex b j is adjacent to only onevertex in [ai ] ∩ Δ and to only one vertex in [ai ] ∩ Δg. �

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By Lemmas 1–3 for all i, j ∈ {1, .., r}, we have |[ai ] ∩ [b j ]| = (|S| − 1)/r . By 2-transitivity of G on Σ , we conclude that any two vertices of Γ from different fibres haveexactly μ = (|S| − 1)/r common neighbours. Thus Γ is an arc-transitive distance-regular graph with intersection array {|S|, (|S| − 1)(r − 1)/r, 1; 1, (|S| − 1)/r, |S|}.Now, we show that Γ admits distance-regular quotients.

Lemma 4 Let H0 = H〈 f t 〉, where t is a proper divisor of r . Then Γ (G, H0, H0gH0)

is an arc-transitive distance-regular graph with intersection array {|S|, (|S| − 1)

(t − 1)/t, 1; 1, (|S| − 1)/t, |S|}.Proof It is clear that 〈 f t 〉-orbit on F(H) containing the vertex H is an imprimitivityblock of the group G on the set of vertices of Γ . Hence the set π = {{H xm(gs)e| x ∈H0}| m ∈ M, s ∈ S, e ∈ {0, 1}} forms an imprimitivity system of G on the verticesof Γ . Moreover, the partition π is equitable. By [4, Theorem 6.2], we have thatΓ ′ = Γ/π is an antipodal distance-regular graph with intersection array {rμ+1, (t −1)rμ/t, 1; 1, rμ/t, rμ + 1}. Since H acts transitively on blocks of π , which containthe neighbours of H in Γ , it follows that G acts transitively on arcs of Γ ′, and byProposition 1 Γ ′ � Γ (G, H0, H0gH0).

As the mapping H y → H y f tis an isomorphism between the graphsΓ (G, H, HgH)

and Γ (G, H, Hg f 2t H) for each t ∈ {1, ..., r}, and 〈 f 〉 = 〈 f 2〉, we see that Γ doesnot depend (up to isomorphism) on the choice of involution g ∈ G − M . The samealso holds for Γ (G, H0, H0gH0). This completes the proof of Theorem 1. �Corollary 1 Let G = Sz(q), where q = 22a+1 > 2. Suppose that Φ is the graphwhose vertices are the involutions of G, and two vertices are adjacent if and only if theorder of the product of corresponding involutions in G is 5. Then Φ is an arc-transitiveantipodal distance-regular graph with intersection array {q2, q2 − q − 2, 1; 1,

q + 1, q2}.Proof Note that the graph Φ is connected. Moreover, G acts transitively (by conju-gation) on arcs of Φ, see [10]. Let x be a vertex of Φ and put S = CG(x). Clearlythe stabilizer of x in G is precisely S, and S ∈ Syl2(G). Let g be an involution of Gsuch that |xg| = 5. Then x and xg are adjacent, and by Proposition 1 Φ � Γ (G, S,

SgS). �Acknowledgments The author thanks the referees for useful comments and suggestions. This work wassupported by the Russian Foundation for Basic Research (Project No. 14-01-31298 MOJI_a) and by a Grantfor young researchers from the Ural Branch of the Russian Academy of Sciences (Project No. 14-1-HII-278).

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