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3URAL MATHEMATICAL JOURNAL, Vol. 8, No. 2, 2022, pp. 162-176
DOI: 10.15826/umj.2022.2.014
ON SOME VERTEX-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS1
Ludmila Yu. Tsiovkina
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation
Abstract: In the present paper, we classify abelian antipodal distance-regular graphs r of diameter 3 with the following property: (*) r has a transitive group of automorphisms G that induces a primitive almost simple permutation group Gs on the set S of its antipodal classes. There are several infinite families of (arc-transitive) examples in the case when the permutation rank rk(Gs) of Gs equals 2; moreover, all such graphs are now known. Here we focus on the case rk(Gs) = 3. Under this condition the socle of Gs turns out to be either a sporadic simple group, or an alternating group, or a simple group of exceptional Lie type, or a classical simple group. Earlier, it was shown that the family of non-bipartite graphs r with the property (*) such that rk(Gs) = 3 and the socle of Gs is a sporadic or an alternating group is finite and limited to a small number of potential examples. The present paper is aimed to study the case of classical simple socle for Gs. We follow a classification scheme that is based on a reduction to minimal quotients of r that inherit the property (*). For each given group Gs with simple classical socle of degree |S| < 2500, we determine potential minimal quotients of r, applying some previously developed techniques for bounding their spectrum and parameters in combination with the classification of primitive rank 3 groups of the corresponding type and associated rank 3 graphs. This allows us to essentially restrict the sets of feasible parameters of r in the case of classical socle for Gs under condition |S| < 2500.
Keywords: Distance-regular graph, Antipodal cover, Abelian cover, Vertex-transitive graph, Rank 3 group.
1. Introduction
Let r be an antipodal distance-regular graph of diameter three. Then r is an antipodal r-cover of a complete graph on k + 1 vertices, and its intersection array has form {k, (r — 1; k}, where k, r and rf denote the valency of r, the size of its antipodal classes and the number of common neighbours for each two vertices at distance two of r, respectively (e.g. see [2]); for brevity, we will refer to such a graph as an (k + 1, r, rf)-cover. We denote by CG(r) the group of all automorphisms of r fixing setwise each of its antipodal classes. If the group CG(r) is abelian and acts regularly on (every) antipodal class of r, then r is called an abelian (k + 1,r, rf)-cover (see [5]). There are some important links between abelian covers and other combinatorial or geometric objects (we refer to [9] and [5] for more background). The problem of finding new their constructions involves many natural questions on possible structure of such a graph, and one of them is to study vertex-transitive representatives.
In the present paper, we classify abelian (k + 1, r, rf)-covers r with the following property:
(*) r has a transitive group of automorphisms G that induces a primitive almost simple permutation group GS on the set £ of its antipodal classes.
Without loss of generality, we may assume tha^ G coincides with the full pre-image of GS in Aut(r). When the permutation rank rk(GS) of GS equals 2, there are several infinite families of
1This work is supported by the Russian Science Foundation under grant no. 20-71-00122.
(arc-transitive) examples; moreover, all such graphs are now known. Here we focus on the case rk(Gs) = 3. Under this condition the socle of Gs turns out to be either a sporadic simple group, or an alternating group, or a simple group of exceptional Lie type, or a classical simple group (see [3, Ch. 11] for an overview on classification of primitive rank 3 permutation groups).
In [16] and [17], it was shown that the family of non-bipartite graphs r with the property (*) such that rk(Gs) = 3 and the socle of Gs is a sporadic or alternating group is finite and limited to a small number of potential examples. The present paper is aimed to study the case of classical simple socle for Gs. We follow a classification scheme that was proposed in [16] and that is based on a reduction to minimal quotients of r that inherit the property (*). For each given group we determine potential minimal quotients of r, applying the constraints for their spectrum and parameters obtained in [16] in combination with the classification of primitive rank 3 groups of the corresponding type (see [8], [11], and also [13]) and associated rank 3 graphs (see [3, Ch. 11]). This allows us to essentially restrict the sets of feasible parameters of r in the case of classical socle for Gs with |£| < 2500. In particular, we show that for most of these sets r must be a covering of a certain distance-transitive Taylor graph.
2. Preliminaries
We keep the notation and terminology from [16] and we refer the reader to [1] and [2] for basic definitions. Next we recall some of them. For a finite group G, we denote by Soc(G), Z(G) and G' its socle, center and derived subgroup, respectively. If G = G', then M(G) denotes its Schur multiplier. If G = 1, then we write "dmin(G)" to denote the number |G : H|, where H is a proper subgroup of G of the smallest possible index. Further, if G is a transitive permutation group on a finite set Q and Orb2(G) is the set of G-orbitals on Q, then the number |Orb2(G)|, denoted by rk(G), is called the (permutation) rank of G. For each Q € Orb2(G), Q* denotes the orbital paired with Q. If Q* = Q and a € Q, then Q(a) denotes the set of all points b € Q such that (a, b) € Q.
In what follows, we consider only undirected graphs without loops or multiple edges. For a graph r by V(r) and A(r) we denote its vertex set and the arc set, respectively. An (n, r, ^)-cover is equivalently defined as a connected graph, whose vertex set admits a partition into n cells (called antipodal classes or fibres) of the same size r > 2 such that each cell induces an r-coclique, the union of any two distinct cells induces a perfect matching, and every two non-adjacent vertices that lie in distinct cells have exactly ^ > 1 common neighbours. Since an (n, r, ^)-cover is bipartite if and only if r = 2 and ^ = n — 2, and for each n > 3 there is a unique (abelian) (n, 2,n — 2)-cover (see [2, Corollary 1.5.4]), we omit these from further consideration. We will say that the set of parameters (n, r, of a non-bipartite abelian (n, r, ^)-cover r is feasible if it satisfies the known necessary conditions for the existence of r that are collected in [16, Proposition 1] (see [16] for detailed references) and [5, Lemma 3.5, Theorem 5.4]. In view of [5], for every (n, r, ^)-cover r and every subgroup N of CG(r) of order less than r, the quotient rN that is defined as the graph on the set of N-orbits in which two vertices are adjacent if and only if there is an edge of r between the corresponding orbits, is a (n, r/|N|,^|N|)-cover. Hence if r is a non-bipartite abelian (n, r, ^)-cover, then, using decomposition CG(r) = Op(CG(r)) x N, where p is a prime divisor of r, we obtain that r possesses a quotient
rN that is a non-bipartite abelian (n,p',^|N|)-cover with p1 = |Op(CG(r))|. Clearly, the factor group Aut(r)/N acts as a group of automorphisms of rN, and in case CG(r) > M > N other quotients rM inherit a similar property when M < Aut(r). Thus parameters of r may depend on the structure of CG(r). This is also demonstrated by the fact that for each non-bipartite abelian (n, r, ^)-cover, every prime divisor of r is also a divisor of n (see [5, Theorem 9.2] and also [6, Theorem 2.5]). These basic observations are crucial for our following arguments; they will be used further without any additional reference.
The next result from [16] distinguishes several types of quotients that an abelian non-bipartite
(k + 1, r, y)-cover with the property (*) may possess.
Proposition 1 [16, Proposition 2]. Let Г be a non-bipartite (k + 1,r, y)-cover and £ be the set of its antipodal classes. Suppose Г has a transitive automorphism group Gi which induces a primitive almost simple permutation group Gis on £ and put T = Soc(Gis). Let G be the full pre-image of the group T in Gi and K be the kernel of the action of the group G on £. Then K contains a subgroup N that is normal in Gi and satisfies one of the following conditions (below the symbol " denotes factorization with respect to N) :
(Tl) К ~ Epi is an elementary abelian group of exponent p and either
(i) G = K xG' and G' ~ T, or (ii) G is a quasi-simple group with center K;
(T2) К ~ Epi is an elementary abelian group of exponent p, T acts faithfully on K, i.e. T < GLi(p), and dmin(T) < (pl - 1)/(p - 1);
(T3) К ~ Sl, where S is a simple non-abelian group, and either
(i) G = K x Cq(K) and С^(К) ~ T, or
(ii) G < Aut.(A") and T contains a proper subgroup of index dividing I.
Each graph r that satisfies the hypothesis of Proposition 1 will be referred to as a minimal (k + 1,r, rf)-cover of type (Tx) with x = 1,2,3 and denoted by r(G1, G, K) if |K| = r, the triple (G1,G, K) satisfies the condition (Tx) from the conclusion of Proposition 1 and K is a minimal normal subgroup of G1 (in particular, N = 1). Thus, for a minimal (k + 1,r, rf)-cover r(G1,G, K) the number r is a prime when G1 = G and K < Z (G).
From now on rjs a non-bipartite abelian (k +1, r, rf)-cover with property (*), £ is the set of its antipodal classes, G is a transitive groupjrf automorphisms of r which induces a primitive almost simple permutation group GS on £, rk(GS) = 3, k1 and k2 are the non-trivial subdegrees of GS, K = CG(r) < G and G is the full pre-image of the group Soc(GS) in G.
Now we proceed with final technical definitions. For a vertex x of r, by F(x) and r1(x) (or [x]) we denote, respectively, the antipodal class of r containing x, and its ^neighborhood in r. Put Q = V(r), and fix a € Q and F = F(a). Let M = G{F} and H = Ga (note |K| = r implies M = K : H). Then A(r) = Q U Q2 for some ^1,^2 € Or^G) with Qi = Q* (see [16]), IQi| = rki(k + 1), and |H : Cia,bi| = ki for each arc (a,bi) € Qi, so H has exactly two orbits on r1(a) (with representatives b1 and b2). For i = 1,2, let denote the (rank 3) graph on £ in which two vertices F(x) and F(y) are adjacent if and only if (x,y) € Qi. If rk(GS) = 3 then the group GS is also primitive as = 0, ki (see, for example, [1, 16.4]). Moreover, the parameters k1, k2
and A satisfy the following equation (see [16])
(A — A1)k1 = (A — A2)k2,
where Ai = |r1(bi) n H(bi)|, i = 1,2. We will say that r admits an H-uniform edge partition (with parameters )) (see [16]), if for each j = 1,2 and for every two distinct vertices z1, z2 € F,
the number of edges between Qj(z1) and Qj (z2) is constant and equal to kjrfj, where rfj is a fixed integer.
Lemma 1 [16, Lemma 1]. Suppose that G{F} = Ga x K and rk(GS) = 3. If H acts transitively on F \ {a} or r < 3, then r admits an H-uniform edge partition.
Theorem 1 [16, Theorem 1]. Suppose that G{F} = Ga x K and rk(Gs) = 3. Then for each x € F \ {a} we have
(y - = (y - ^2)^2,
where yi = |r1(bi) n Qi(x)|, i = 1,2. If, moreover, r admits an H-uniform edge partition with parameters (y'^y^), then yi = yi (in particular, ki — 1 = Ai + (r — 1)^i) for every i = 1,2 and Y = —(A — A1 — A2) + (y — y1 — y2) = r(y — y1 — y2) — 1 is an eigenvalue of r.
3. Main results
Theorem 2. Suppose that r = r(G,G, K) is a minimal abelian (k+1, r, y)-cover, k+1 < 2500, rk(Gs) = 3 and T = Soc(Gs) is a classical simple group, isomorphic to the group M/Z(M), where M = Sp2n-2(q), Q±n(q), ^2n-1(q) or (q) for n > 3. Assume G = G whenever rk(T) = 3. Then one of the following statements is true :
(1) T ~ PSU4(4), rk(T) = 3, k + 1 = 1105, r = 5 and y = 210;
(2) T ~ G' ~ P^±ra(2), rk(T) =3, k + 1 = (22n-1 — e2n-1), where e = ±1 and n < 6, 2(A($1) + A($2) + 1) = k — 1, r = 2 and either G = G' ~ Z2.PQ+ (2), e = +1, k + 1 = 120, and y € {64, 54}, or the group G' is intransitive on V(r);
(3) T ~ G' ~ P^5(8) ^ PSp4(8), rk(T) = 5, 2(A($1 ) + A($2) + 1)= k — 1, k + 1 = 2016 and ry € {2048,1980}, or k + 1 = 2080 and ry € {2048,2108}, wherein either r = 4 and G' is intransitive on V(r), or r = 2 and G' is transitive on V(r);
(4) T ~ POm(q), rk(T) = 3, 2(A($0 + A($2) + 1) = k — 1, r = 2 and either
(i) m = 5, q = 3, k + 1 = 36 and y € {16,18}, or
(ii) m = 5, q = 4, with k + 1 = 120 and y € {54, 64} or k + 1 = 136 and y € {64, 70}, or
(iii) m = 7, q = 4, with k + 1 = 2016 and y € {990,1024} or k + 1 = 2080 and y € {1024,1054},
and in all cases (i) —(iii) the group G' is intransitive on V(r);
(5) T ~ G' ~ SUs(3), rk(T) = 4, k + 1 = 36, 2(A($1 ) + A($2) + 1) = k — 1, r = 2, y € {16,18} and G' is intransitive on V (r);
(6) T ~ G' ~ PSpe(2) ^ Pfir(2), rk(T) = 3, k + 1 = 120, 2(A($0 + A($2) + 1) = k — 1, r = 2, y € {54,64} and G' is intransitive on V(r).
Moreover, if r = 2 and G' ~ T, then for any given pair of parameters k and y, r is a unique (up to isomorphism) distance-transitive (k + 1,2, y)-cover.
Proof. Let k + 1 < 2500. Under this condition rk(T) = 3 for all k except the following cases (a)-(d) (note that in [16, Example] the case (d) is missing, and the subdegrees k1,k2 for the case (c) are mistyped):
(a) k + 1 = 36, k1 = 14, k2 = 21, T ~ PSL2(8), rk(T) = 5, (5s ~ P^(8) = T.3;
(b) k + 1 = 36, k1 = 14, k2 = 21, T ~ PSUs(3), rk(T) =4, Gs ~ PrUs(3) = T.2;
(c) k + 1 = 2016, k1 = 455, k2 = 1560, Gs ~ 6^4(8), rk(Gs) = 5 and Gs ~ Sp4(8).Zs
(d) k + 1 = 2080, ki = 567, k2 = 1512, Gs — Sp4(8), rk(Gs) = 5 and Gs — Sp4(8).Zs.
Then, by [16, Propositions 2, 3], if rk(T) =3, T < Aut(K) and 2(A($i) + A($2) + 1) = k — 1, then either G' — T acts transitively on V(r), or G is a quasisimple group. Therefore, in order to find some necessary conditions for r to exist (as well as for its covers with property (*)), in case rk(T) = 3 it suffices to consider the case G = G, and if, moreover, K < Z(G), then one may assume that r is prime. Taking this into account, we further specify the possible structure of G for each potential pair (Gs, ).
Throughout the rest of the proof, we put N = G' and denote by d and — t, respectively, the positive and negative eigenvalues of r, other than k and —1. We will consider the following possible combinations for T and complementary rank 3 graphs and $2 associated with Gs, applying their description from [8] and [3, Theorem 11.3.2].
(A) Let k1 = q(qn-1 — 1)(tqn-1 + 1)/(q — 1) and suppose the graph $1 has parameters
qn _ 1 qn-1 — 1 «"-2 — i qn-1 — 1
(--tW1 + !)> Q---^W2 + 1), Q--tqn~3 + !) + <?-!, --tqn~2 + 1)),
v q — 1 q — 1 q — 1 q — 1 y
where i = q, 1, q, q2, q1/2, q^2 for M = Sp2n(q), ^(9), iWi(i), Q2n+2(<?), SU2n(y/q) or respectively (see [3, Theorem 11.3.2(i)]).
By condition k + 1 < 2500, hence the equality 2(A($1) + A($2) + 1) = k — 1 holds if and only if t = 1, q = 3, n = 2 and (v,k1 ,A($1),y($1)) = (16,6,2,2), which contradicts the constraint n > 3 for t = 1. If r is a power of a prime p, say r = p1, then feasible sets of parameters k, r, and y are described by Table 1, and r has no H-uniform edge partitions in the cases t = 1, q, q2 (this can be easily checked by complete enumeration in GAP [14], based on Theorem 1, [16, Proposition 1] and [5, Lemma 3.5, Theorem 5.4]).
(A1) Let T ~ PSp2n(q) and k + 1 = (q2n — 1)/(q — 1). Then rk(T) = 3, while dmin(T) = k + 1, except for the cases when q = 2, 2n > 6 and dmin(T) = 2n-1 (2n — 1) or 2n = 4, q = 3 and dmin(T) = 27 (see [12, Theorem 2]). Moreover, m(t) = Zgcd(2;q-1) for (q; n) = (2; 2), (2;3) and M(T) = Z2 for (q; n) = (2; 2), (2; 3), Out(T) = Zgcd(2>q-1) ■ Ze, where q = pe, p is a prime.
According to Table 1 (q; n) € {(2; 3), (3; 2)}. Hence dmin(T) = k + 1. It follows that K < Z(G) and, as noted above, it suffices to consider the case of prime r.
Since r has no H-uniform edge partitions, we have r > 5. Also, 2(A($1) + A($2) + 1) = k — 1. Hence, due to [16, Proposition 3] N = G' ~ T acts transitively on V(r). But then Ga — N{F} contains a subgroup of index r and G{F} = GaN{F}.
If n = 3 = q, then (|N|)5 = 5 and hence |N{F}| is not divisible by 5, a contradiction. Let n = 2. Then N{F} is an extension of a group of order q3 by a group of the form ((q — 1)/2 x L2(q)).2 or ((q — 1) ■ L2(q)) (see, for example, [4] or [13]). In any case, N{F} does not contain subgroups of index 5, a contradiction.
(A2) Let T — O2n+1(q), t = q and k + 1 = (q2n — 1)/(q — 1). Recall that PSp4(q) — O5(q) for n = 2, and also that O2n+1(q) — PSp2n(q) for even q (see, for example, [18]). Since the corresponding cases are considered in case (A1), we will further assume that n > 3 and q is odd. Then rk(T) = 3 and by [18, Theorem] dmin(T) = k + 1, except for the case q = 3, in which dmin(T) = 3n(3n — 1)/2. Moreover, M(T) = Z(2>q-1) for (q; n) = (3;3) and M(T) = Z2 x Z2 x Z3 for q = 3 = n (e.g. see [7]).
As in case (A1) we have q = n = 3 and since dmin(T) > r we conclude K < Z(G). Hence we may assume that r is prime. But then Table 1 gives r = 2 and hence by Lemma 1 and Theorem 1 r admits an H-uniform edge partition, a contradiction.
Table 1. Feasible parameters of r with r = pl in case (A)
q, n k + 1 h, h e —r r/i r
Type t = q:
7, 2 400 56, 343 19 -21 400 2,4,5,8,25
21 -19 396 2
9, 2 820 90, 729 21 -39 836 2
39 -21 800 2,4,5,8,16,25
3, 3 364 120, 243 11 -33 384 2,4,8,16
33 -11 340 2
Type t = 1:
Type t = q2:
4, 2 325 68, 256 9 -36 350 5
12 -27 338 13
3, 3 1066 336, 729 ^1065 -^1065 1064 2,4
2, 4 495 238, 256 19 -26 500 5,25
26 -19 486 3,9,27,81,243
Type t = ^Jq:
4, 2 45 12, 32 4 -11 50 5
11 -4 36 3,9
9, 2 280 36, 243 9 -31 300 2,5
31 -9 256 2,4,8,16,32,64,128
16, 2 1105 80, 1024 16 -69 1156 17
69 -16 1050 5,25
Type t = ^/q6: 0
(A3) Let T ~ O+n(q), where n > 3, t = 1 and k +1 = (qn - 1)(qn-1 + 1)/(q - 1). Then condition k + 1 < 2500 implies either n = 3 and q < 5, or n = 4 and q < 3, or n = 5,6 and q = 2. According to Table 1 none of these cases is possible.
(A4) Let T ~ O-n(q), where n > 2, t = q2 and k + 1 = (qn - 1)(qn+1 + 1)/(q - 1). Then rk(T) = 3 and in view of Table 1 n = 2,3,4. Recall that O-(q) ~ L2(q2) and O-(q) ~ U4(q) (e.g. see [18]).
If n = 4 and q = 2, then by [4] dmin(T) = 119. By [12, Theorem 1, Theorem 3] dmin(T) = q2 + 1 = 17 for n = 2 and dmin(T) = (q3 + 1)(q + 1) = 112 for n = 3 = q. In each case dmin(T) > r and hence K < Z(G). Arguing as in case (A3), we obtain that r = 5 and N = G acts transitively on V(r). But then N ~ L2(16) or O-(2), and |N| is not divisible by 25. This contradicts the fact that N{F} must contain a subgroup of index r.
(A5) Let T ~ PSU2n(y/q)- In view of Table 1 either T ~ PSUA{2) ~ PSpA{3) and dmin(T) = 27 or T ~ PSU4(3) ~ O-(3) < GLz(2) and dmin(T) = 112, or T ~ PSU4(4) and dmin(T) = 325 (see [4] and [12, Theorem 3]). Hence K < Z(G) and we may assume that r is a prime. If G is a quasi-simple group, then r divides |M(T)| and so by [7] r = 2 and q = 9. If N ~ T acts transitively on V(r), then r2 divides |N| and so r = 5 for q = 16 and r < 3 for q < 9.
Suppose q = 9. Then r = 2, r admits an H-uniform edge partition with parameters ^2) and |(A1, A2), (^1,^2)} = {(15,130), (20,112)}. Enumeration of orbital graphs in GAP [14] shows that r does not exist when N ~ T. But for G = N the groups (Ga)[a] and (G{F}} are permutation isomorphic. Moreover, for the vertex b1 € Q1(a) the group Ga,bl has exactly two orbits of length 4 and one orbit of length 27 on [a], which contradicts the fact A1 € {15,20}.
Suppose q = 4. Then r = 3, N ~ T acts transitively on V(r), (Ai, A2) = (3,13) and r admits an H-uniform edge partition with parameters (y1,y2) = (4,9). A complete enumeration of orbital graphs in GAP [14] shows that this case cannot occur.
(A6) In the case of T ~ PSU-2n+i(\fq) and t = >/q3 we have n = 2 and q = 4,9, but according to Table 1 none of the cases gives a feasible parameter set.
(B) Let us consider the cases T ~ M/Z(M), where M = Sp4(q), SU4(q), SU5(q), ^-(q), (q) or n+0(q) from [3, Theorem 11.3.2(ii)] (see also [8]).
(B1) Let k1 = t(q + 1) and the graph $1 have parameters
((t + 1)(tq + 1),t(q + 1), t - 1, q + 1),
where t = q, q2, q1/2, q3/2 for M = Sp4(q), f2g (q), SU^^Jq) or SU^i^/q), respectively. If r = pl is a power of a prime p, then feasible sets of parameters k, r, and y are described by Table 2, and r does not admit, ii-uniform edge partitions when t = q, sjq (this can be easily checked in GAP [14], applying Theorem 1, [16, Proposition 1] and [5, Lemma 3.5, Theorem 5.4]). Moreover, cases t = q, q2, sjq correspond to the above cases (Al), (A5) and (A4), respectively.
Table 2. Feasible parameters of r with r = pl in case (B)
Q k + 1 h, h e —r r/i r
(Bl), type t = = Q-
7 400 56, 343 19 -21 400 2,4,5,8,25
21 -19 396 2
9 820 90, 729 21 -39 836 2
39 -21 800 2,4,5,8,16,25
(Bl), type t - = Q2-.
2 45 12, 32 4 -11 50 5
11 -4 36 3,9
3 280 36, 243 9 -31 300 2,5
31 -9 256 2,4,8,16,32,64,128
4 1105 80, 1024 16 -69 1156 17
69 -16 1050 5,25
(Bl), type t - = VQ-
16 325 68, 256 9 -36 350 5
12 -27 338 13
(Bl), type t - 0
(B2),(B3): 0
(B2)&(B3) Let T = ki = q(q2 + 1)(q3 - 1)/(q - 1) and the graph have parameters
(1 + Q(Q2 + 1)^ + <?6, Q(Q2 + + " " 1. (I2 + 1)^).
(see [3, Theorem 2.2.17, Proposition 3.2.3]) or let T = Q+0(q)> ki = q(q2 + 1)(q5 - 1)/(q - 1) and the graph $1 have parameters
q5 — 1
((q4 + l)(q3 + 1 )(q2 + l)(q + 1), q(q2 + 1)^--,q - 1 + q2(q + 1 )(q2 + q + 1), (q2 + l)(q2 + q + 1)).
q - 1
As k + 1 < 2500, it follows that q < 3 and hence either k + 1 = 135,1120 and T = O+(q) for q = 2, 3 respectively, or q = 2, k + 1 = 2295 and T = O+0(2). According to Table 2 in any case, none of the parameter sets k, r and ^ is feasible (this was checked in GAP [14] using [16, Proposition 1]) and [5, Lemma 3.5, Theorem 5.4]).
(C) Now let us consider the cases T ~ M/Z(M), where M = SUm(2), ^±m(2), ^±m(3), fi2m-1(3), Q2m-1(4) or Q2m-1(8) for m > 3, from [3, Theorem 11.3.2 (iii,iv)] (see also [8]).
(C1) Let T = Un(2) (see [3, § 3.1.6]) and the graph $ = NUn(2) have parameters
(2n-1(2n - e)/3, (2n-1 + e)(2n-2 - e), 22n-53 - e2n-2 - 2,2n-33(2n-2 - e)), where e = (- 1)n.
In view of Table 3 we have n = 5 and k + 1 = 176, i.e. T ~ U5(2). Since
2(A($1) + A($2) + 1) = k - 1
and r divides 4, then either N ~ T acts transitively on V(r), or G is a quasisimple group and by [7] K < M(T) = Z2. But in the first case, by [4], L = N{F} ~ Z3 x U4(2) has no subgroups of index r, a contradiction. In the second case r = 2 and r admits an H-uniform edge partition with parameters (^1,^2), and {(^1,^2), (A1 ,A2)} = {(78,21), (56,18)}. But then subdegrees of the group Ga on Q1(a) (recall that |Q1 (a) | — k1) are as follows: 11, 61, 324, 361 (the upper indices denote the multiplicities of the corresponding subdegrees). This contradicts the fact A1 € {78, 56}.
(C2) Let T = Pfi±n(2) (see [3, § 3.1.2]) and the graph $1 = NO|n(2) have parameters
^2n-1_e2n-1 22n-2_1 22n-3_2 22n-3 + e2n-2)
where e = ±1. Since k + 1 < 2500, n < 6. Then
2(A($1) + A($2) + 1) = k - 1
for all n and e (see also [17, Example 1]). Suppose n = 3.
If T ~ PQ+(2) ~ L4(2) ~ Altg, then r = 2 and N is intransitive on V(r) (note that r is a graph from [17, Theorem 2(ii)]).
Let T ~ PQ-(2) ~ ^4(2) ~ PSp4(3). Then k + 1 = 36, M(T) = Z2 and rk(T) = 3. Since dmin(U4(2)) = 27 (see [4]), we get K < Z(G).
Assume that N is transitive on V(r). Then r = 2, N = G ~ Sp4(3) or PSp4(3). Consequently, Ga ~ SL2(9) or Ga ~ Alt6. In the first case K = Z(G) < Ga, and in the second case the rank of the transitive representation N on V(r) is equal to 5. Both cases are impossible.
Let n > 3. Since dmin(T) = 2n-1 (2n - 1) (see [18]) for e = +1, dmin(T) = 119 (see [4]) for e = -1 and n = 4, dmin(T) = 495 (see [4]) for e = -1 and n = 5, and dmin(T) = 2015 (see [13]) for e = -1 and n = 6, we get K < Z(G). Then, by [16, Proposition 3], either N ~ T is intransitive on V(r), or N is transitive on V(r). Let us consider the second case. Recall that M(T) = Z2 x Z2 for n = 4, e = +1 and M(T) = 1 otherwise (e.g. see [7]). Further, the group T{F} is isomorphic to the group PSp2n(2) (see [13]) and it has no subgroup of index r from the corresponding case in Table 3. Hence N = G, n = 4, e = +1 and r = 2. By Lemma 1 and Theorem 1 r admits an H-uniform edge partition with parameters (^1,^2) and {(A1,A2), (^1,^2)} = {(32,28), (30,27)}, and ^ € {64,54}.
Table 3. Feasible parameters of r with r = pl in cases (C1)-(C3)
n k + 1 hi, k2 e —r r/J r
(CI) 5 176 135, 40 5 -35 204 2
35 -5 144 2,4
(C2), e = —1: 3 36 15, 20 5 -7 36 2,3,9
7 -5 32 2,4,8,16
4 136 63, 72 9 -15 140 2
15 -9 128 2,4,8,16,32,64
5 528 255, 272 17 -31 540 2,3,9,27
31 -17 512 2,4,8,16,32,64,128,256
6 2080 1023, 1056 9 -231 2300 2
21 -99 2156 2
27 -77 2128 2,4,8
33 -63 2108 2
63 -33 2048 r = 2\l < 10
77 -27 2028 2,13,169
99 -21 2000 2,4,5,8,25,125
231 -9 1856 2,4,8,32
(C2), e = +1: 3 28 15, 12 3 -9 32 2,4
9 -3 20 2
4 120 63, 56 7 -17 128 2,4,8
17 -7 108 2,3,9,27
5 496 255, 240 15 -33 512 2,4,8,16
33 -15 476 2
6 2016 1023, 992 13 -155 2156 2,7
31 -65 2048 2,4,8,16,32
65 -31 1980 2,3,9
155 -13 1872 2,3,4
(C3), e = -1: 3 126 45, 80 5 -25 144 2,3
25 -5 104 2,4
— >/125 124 2
(C3), e= 1: 0
Table 4. Feasible parameters of r with r = pl in case (C4)
e, q, n k + 1 ki, k2 e —r rfi r
(C4) -1,3,2 36 20, 15 5 -7 36 2,3,9
7 -5 32 2,4,8,16
-1,3,3 351 224, 126 14 -25 360 3,9
25 -14 338 13,169
35 -10 324 3,9,27,81
1,3,2 45 32, 12 4 -11 50 5
11 -4 36 3,9
1,3,3 378 260, 117 13 -29 392 2,7
29 -13 360 2,3,4,9
>/377 -y/377 376 2,4
-1,4,2 120 51, 68 7 -17 128 2,4,8
17 -7 108 2,3,9,27
-1,4,3 2016 975, 1040 13 -155 2156 2,7
31 -65 2048 2,4,8,16,32
65 -31 1980 2,3,9
155 -13 1872 2,3,4
1,4,2 136 75, 60 9 -15 140 2
15 -9 128 2,4,8,16,32,64
1,4,3 2080 1071, 1008 9 -231 2300 2
21 -99 2156 2
27 -77 2128 2,4,8
33 -63 2108 2
63 -33 2048 r = 2\l < 10
77 -27 2028 2,13,169
99 -21 2000 2,4,5,8,25,125
231 -9 1856 2,4,8,32
-1,8,2 2016 455, 1560 13 -155 2156 2,7
31 -65 2048 2,4,8,16,32
65 -31 1980 2,3,9
155 -13 1872 2,3,4
1,8,2 2080 567, 1512 9 -231 2300 2
21 -99 2156 2
27 -77 2128 2,4,8
33 -63 2108 2
63 -33 2048 r = 2\l < 10
77 -27 2028 2,13,169
99 -21 2000 2,4,5,8,25,125
231 -9 1856 2,4,8,32
A computer check in GAP [14] shows that in the case when r = 2, N ~ T and N is intransitive on V(r), r exists and it is unique distance-transitive (k + 1,2,y)-cover (note it can be also constructed using [17, Theorem 1] or appears in [17, Example 1]).
(C3) Let T = Pfi±n(3) (see [3, § 3.1.3]) and the graph $1 = NO|n(3) have parameters
(I3"-l(3" _ £)} I3™-1(3"-1 _ £)} I3™-2(3™-l + t-)) l3n-l(3n-2 _ ^
where e = ±1.
In view of Table 3 we have k +1 = 126, e = -1 and r < 4. Then T ~ U4(3) and dmin(T) = 112 (see [4]). Hence K < Z(G). Enumeration of feasible parameters in GAP [14] shows that r does not admit H-uniform edge partitions when A = y, a contradiction with Lemma 1 and Theorem 1.
If N ~ T acts transitively on V(r), then N{F} ~ U4 (2) contains a subgroup of index r < 4, a contradiction. Therefore G = N is a quasi-simple group and, by [7], r = 2. Hence, by Lemma 1 and Theorem 1, r admits an H-uniform edge partition with parameters (y1,y2) and |(Ai,A2), (^1,^2)} = {(24,45), (20,34)}. Since G = N, the groups (Ga)[a] and (G{F})S-{F} are permutation isomorphic. Moreover, for the vertex b1 € Q1(a) the group Ga,b1 has exactly two nonsingle-point orbits on Q1(a): one orbit of length 12 and one orbit of length 32. This is impossible, since A1 € {20, 24}.
(C4) Let T = PH2n+1(q) (see [3, § 3.1.4]) and the graph $1 = NO2n+1 (q) have parameters
{\(ln{(ln + e), (qn~l + e)(qn - e), 2{q2n~2 - 1) + eqn~\q - 1), 2qn~l{qn~l + ej),
where e = ±1, q = 3,4,8 and n > 2. According to Table 4, the equality 2(A($1) + A($2) + 1) = k-1 holds only when either k + 1 = 36 and q = 3 or q = 4.
For q = 3 we have either n = 2 and dmin(T) = 27, or n = 3 and dmin(T) = 351 (see [4]). For even q we have PQ2n+1(q) ~ PSp2n(q) and, by [12, Theorem 2], dmin(T) = (q2n — 1)/(q — 1), i.e. dmin(T) = 85 for 2n = q = 4, dmin(T) = 585 for 4n = q = 8 and dmin(T) = 1365 for n = 3 and q = 4. Moreover, r = pl > dmin(T) is possible only for 4n = q = 8. Together with the fact that PSp4(8) < GL1o(2), this implies K < Z(G).
First we consider the cases when 2(A($1) + A($2) + 1) = k — 1.
If e = +1, q = 3 and n = 2 then T ~ P^(3) ~ PSp4(3) and rk(T) = 3. This possibility was treated in case (A5).
Let q = n = 3. Then T ~ P^(3), rk(T) = 3 and k + 1 is equal to 351 (for e = —1) or 378 (for e = +1). In any case by [4] L has no subgroup of index 3, 7 or 13.
Hence if N ~ T is transitive on V(r) then r = 2, e = +1 and Na = NF ~ L4(3) has two orbits on [a]. Moreover, for the vertex b2 € Q2(a) the group Nab2 has exactly two non-single-point orbits on Q2(a) (recall that k2 = |Q2(a)| = 117), and the lengths of these orbits are 80 and 36. This contradicts the fact that by Lemma 1 and Theorem 1 r admits an H-uniform edge partition with parameters (y1,y2) and {(A1,A2), (y1 ,y2)} = {(133, 56), (126,60)}.
Hence G = N and, by [7], M(T) = Z2 x Z3, which together with Table 4 implies r < 3 for k + 1 = 378 and r = 3 for k +1 = 351. Then, by Lemma 1 and Theorem 1, r admits an H-uniform edge partition with parameters (y1,y2). More precisely, if k + 1 = 378, then {(A1, A2), (y1,y2)} = {(133, 56), (126,60)} for r = 2 and {(A1 ,A2), (^1 ,^2)} = {(84,40), (91,36)} for r = 3, and if k + 1 = 351, then {(A^), (^1,^2)} = {(75,40), (73,45)} and r = 3. Since the groups (Ga)[a] and (G{F})S-{F} are permutation isomorphic, in the case r = 2 a contradiction is achieved in a similar way as above. Let r = 3. For k + 1 = 351 the group Ga,bl, where b1 € Q1(a), has five orbits on Q1(a) (recall that k1 = |Q1(a)| = 224): two orbits of length 81, one orbit of length 60 and two single-point orbits. This is impossible, since A1 = 73 or 75. Let k + 1 = 378. Since for
the vertex b2 € Q2 (a) the group Ga,b2 has exactly two non-single-point orbits on Q2(a) (recall that k2 = |Q2(a) | = 117), and the lengths of these orbits are 80 and 36, then A2 = 36. But then ¡2 = 40, which is impossible, since Ga = GF and the group Ga,b2 moves 36 or 80 vertices from Q2(a*) n [b2] for some vertex a* € F(a).
Let q = 8. According to Table 4 T ~ PQ5(8) ~ PSp4(8) and as noted above rk(T) = 5.
Further, the group (G{F}) has the form ¿2(64)Z.Z2 for k + 1 = 2016 and (¿2(8) x L2(8))Z for k + 1 = 2080. Hence, by [16, Proposition 3] and taking into account that M(T) = 1, we obtain either r = 4, one of -65 or 63 is an eigenvalue of r and N is intransitive on V(r), or N ~ T acts transitively on V(r). Let us consider the second case. If k + 1 = 2080 then for a subgroup of index r in N{f} we have either p = 3 and r divides 35, or r = p = 2. If k + 1 = 2016 then for a subgroup of index r in N{F} we have r = p < 3. Enumeration of the orbital graphs of N in GAP [14] shows that the case r = 3 is impossible, while for r = 2 the graph r exists: for k + 1 = 2016 the parameter i equals to 1024 or 990, and for k + 1 = 2080 the parameter i equals to 1024 or 1054. More precisely, for each feasible set of parameters k, it turns out to be the unique (up to isomorphism) distance-transitive (k + 1,2,i)-cover.
Now let 2(A($i) + A($2) + 1) = k - 1.
Let us consider the case when N is transitive on V(r).
For transitive N, the case e = -1, q = 3 and n = 2 was excluded earlier in (C2).
Let q = 4. Then rk(T) = 3 and by [7] M(T) = 1. If n = 2 then by [4] N{F} ~ ¿2(16) (for k + 1 = 120) or (Alt5 x Alt5) : Z2 (for k + 1 = 136) has no subgroup of index 3, so r = 2. If n = 3, then N{F} ~ Pn|(4) : Z2 (see [13]) has no subgroup of index 3, 5, 7 or 13, so r = 2 again. Enumeration of the orbital graphs of PSp2n(q) on r(k + 1) points in GAP [14] shows that none of these cases is realized.
A computer check in GAP [14] shows that in the case when r = 2, N ~ T and N is intransitive on V(r), r exists and it is unique distance-transitive (k +1,2, ¡)-cover (note it can be also constructed using [17, Theorem 1]).
(D) Finally, let the pair (M, Y), where Y is the pre-image in M of a point stabilizer in T, be one of the following (up to conjugacy in Aut(M) (see [3, § 11.3.2, Theorem 11.3.2(v)-(x)])):
(SUs(3),PSLs(2)), (SU3(5),3.Altr), (SU4(3),4.PSL3(4)), (Spe(2),G2(2)), (Qz(3),G2(3)),
(SUa(2), 3.PSU4(3).2);
let further the graph $1 have parameters
(36,14, 4, 6), (50, 7, 0,1), (162, 56,10, 24), (120, 56, 28, 24), (1080, 351,126,108)
or (1408,567, 246, 216),
respectively (for their detailed description, see [3, §-§ 10.14, 10.19, 10.48, 10.39, 10.78, 10.81]). Then feasible parameters of r are described by Table 5, which, in particular, shows the cases k + 1 = 56, 1080 are impossible.
Let T ~ SU3(3). Then rk(T) = 4, M(T) = 1, and by [4] dmin(T) = 28 > r. Hence K < Z(G) and N ~ T. Suppose N is intransitive on V(r). Then by [16, Proposition 3] we have either r = 4 and 7 is an eigenvalue of r, or r = 2 and 7 = —2(A($j) + kj¡($j)/kj + 1) + k is an eigenvalue of r. In the second case 7 € {±7}, which in view of Table 5 implies i € {16,18}. Computer check in GAP [14] shows that for r = 2 and each r exists and it is the only (up to isomorphism) distance-transitive (36,2, ¡)-cover.
Suppose N ~ T is transitive on V(r). Then N{F} ~ L3(2) must contain a subgroup of index r. But in view of [4] the index of a proper subgroup in L3(2) must be divisible by 7 or 8, which
implies r = 8. Enumeration of the orbital graphs of the group SU3(3) on 36r points in GAP [14] shows that this is impossible.
For r = 4 enumeration of the orbital graphs of the group K x SU3 (3) on 144 points in GAP [14] shows this case is also impossible.
In all other cases rk(T) = 3 and dmin(T) > r. Hence K < Z(G) and, by the remark after Proposition 1, we will assume that r is prime.
For T ~ PSU3(5) we have 2(A($i) + A($2) + 1) = k - 1 and by [7] M(T) = Z3. In view of Table 5 r = 2 and hence N = G' ~ T. Enumeration of the orbital graphs of the group Z2 x SU3(5) on 100 points in GAP [14] shows this case is impossible.
For T ~ Spa(2), we have 2(A($i) + A($2) + 1) = k - 1 and, by [7] M(T) = 1, so N = G' ~ T. Since the rank of the representation of the group Sp6(2) on cosets by its subgroup isomorphic to the group G2(2)', equals 5, we obtain that N is intransitive on V(r). Further, in view of Lemma 1 and Theorem 1 r admits an H-uniform edge partition with parameters ^2) and either {(Ai, A2), (^1 ,^2)} = {(28,32), (27,30)} and r = 2, or {(Ai,A2), (^1 ,^2)} = {(18,20), (19,22)} and r = 3. Since the groups (Ga)[a] and (G{F})S-{F} are permutation isomorphic, for b1 e Q1(a) Ga,b1 -orbits on Q1 (a) have lengths 1, 1, 27 and 27. For r = 3 this is impossible, since A1 = 18 or 19. Hence r = 2. Enumeration of the orbital graphs of the group Zr x Sp6(2) on 240 points in GAP [14] shows that r exists and it is distance-transitive with ^ = 54 or 64.
For T ~ PSUe(2) we have 2(A($1) + A($2) + 1) = k - 1 and, by [7], M(T) = Z3 x Z2 x Z2. Since the rank of transitive representation of PSU6(2) on its right X-cosets with X ~ U4(3) equals 5, then G is a quasi-simple group and r = 2. In view of Lemma 1 and Theorem 1 r admits an H-uniform edge partition with parameters (^1,^2) and {(A1,A2), (^1,^2)} = {(286,429), (280,410)}. Since the groups (Ga)[a] and (G{F})S-{F} are permutation isomorphic, for b1 e Q1(a) Ga,bl-orbits on Q1(a) have lengths 1, 320, 30, 96 and 120. This is a contradiction, since A1 = 286 or 280.
Table 5. Feasible parameters of r with r = pl in case (D)
(M, Y) k + 1 kl, h 9 —r r/i r
№(3),PSL3(2)) 36 14, 21 5 -7 36 2,3,9
7 -5 32 2,4,8,16
(SU3( 5),3.Alt7) 50 7, 42 7 -7 48 2,4,8
(Spe(2), G2(2)) 120 56, 63 7 -17 128 2,4,8
17 -7 108 2,3,9,27
(SU6(2),3.PSU4(3).2) 1408 567, 840 21 -67 1452 2,11
67 -21 1360 2,4,8
□
Theorem 3. Suppose that r = r(G, G, K) is a minimal abelian (k+1, r, ^)-cover, k+1 < 2500, rk((5s) = 3 and T = Soc(Gs) ~ PSLd(q)- Assume G = G whenever rk(T) = 3. Suppose further that (T, k + 1) = (Alts, Q)). Then Gs ~ PrL2(8), k + 1 = 36, r = 2, ^ e {16,18}, G' ~ T, G' is transitive on V(r), and r is a unique (up to isomorphism) distance-transitive (36,2, y)-cover.
Proof. Let T ~ PSLd(q). Next we consider potential combinations for T and the complementary rank 3 graphs $1 and $2 associated with (5s, applying their description from [8] and [3, Theorem 11.3.3]. Since k + 1 < 2500, we are left with the following two cases (E) and (H).
(E) Let either T = PSL2(4) ^ PSL2(5) ^ Alt5, k + 1 = (5), or T = PSL2(9) ~ Alto, k + 1 = (2), or T = P6X4(2) ~ Alts, k + 1 = (8), or G = PrL2(8), k + 1 = (9) (see [8] and also [3, Theorem 11.3.3(ii)]). Then $1 ~ T(m) and m = 5,6,8,9, respectively. The cases m < 8 were considered in [17, Theorem 2]. Below we treat the remaining case m = 9.
Let k + 1 = 36 and Gs ~ PrL2(8). Then T ~ L2(8), rk(T) = 4, M(T) = 1, the graph $1 has parameters (36,14, 7,4) and 2(A($1) + A($2) + 1) = k — 1. If r = pl, p is prime, then p < 3. Note that ¿2(8) < GLi(3) for l < 4 and ¿2(8) < GLi(2) for l < 5. Hence K < Z(G). By [16, Proposition 3] r = 3 and if r < 16, then by [16, Proposition 3] G' ~ T is transitive on V(r), which, in view of [4], implies r = 2. Enumeration of the orbital graphs of the group Z2 x L2(8) on 72 points in GAP [14] shows that y = 16 or 18, and r is a unique distance-transitive (36,2,y)-cover (see also [16, Example]).
(H) If T = PSL3(4), T{F} ~ Alt6 and $1 is the Gewirtz graph (with parameters (56,10,0,2)) or T = PSL4(3), T{f} ~ PSp4(3) and $1 ~ NO+(3) (with parameters (117,36,15,9)), then there is no feasible set of parameters. □
Remark 1. In proofs of Theorems 2 and 3, in a computer search for distance-regular orbital graphs we used GAP packages GRAPE [15] and coco2p [10].
Remark 2. An explicit construction of covers with r = 2 and intransitive group G' from the conclusions of Theorem 2 can be found in [17, Theorem 1, Example 1].
Corollary 1. Suppose that ^ is a non-bipartite abelian (n, r', y')-cover with a transitive group of automorphisms X that induces a primitive almost simple permutation group X" on the set 2 of its antipodal classes such that rk(X") = 3 and the pair (X", n) satisfies conditions of Theorem 2 or 3. Then ^ has a minimal quotient r(G,G,K) that is an (n, r, y)-cover from the conclusion of the respective theorem with Soc(X") ~ G/K and r'y' = ry.
4. Concluding remarks
In this paper, we continued studying abelian antipodal distance-regular graphs r of diameter 3 with the property (*): r has^ a transitive group of automorphisms G that induces a primitive almost simple permutation group Gs on the set £ of its antipodal classes. As in [16], we focused on the case rk(Gs) = 3. In [16] and [17], it was shown that in the alternating and sporadic cases for Gs the family of non-bipartite graphs r with the property (*) and rk(Gs) = 3 is finite and limited to a small number of potential examples with |£| € {10,28,120,176,3510}. Here we assumed that the socle of Gs is a classical simple group. The case of classical simple socle seems to be both most interesting and complicated, since, on one hand, there is an infinite family of non-bipartite representative^ r (see [17, Example 1]), and on the other hand, its study requires a profound inspection of Gs. So we started classification of graphs r with "small" |£|. In order to describe minimal quotients of r, we used the technique for bounding their spectrum that is based on analysis of their local properties and the structure of Gs, which was developed in [16] and applied in [16] and [17] for the cases of sporadic, alternating and exceptional socle (the latter was investigated under condition |£| < 2500). As a result, we significantly refined the sets of feasible parameters of r with |£| < 2500 in the case of classical socle, showing, in particular, that for most of these sets r must be a covering of a certain distance-transitive Taylor graph.
We also wish to mention two more challenging examples of graphs with the property (*), namely, abelian (n, 3,12)-covers with n = 36 or 45 and rk(Gs) = 4 or 5, respectively (for their constructions,
see [9]). A computer assisted inspection shows that they are the only minimal abelian (n, recovers r(G,G, K) such that 3 < rk(Gs) < 5, r > 2, n < 2500 and G = G' is a quasi-simple group.
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