The uniqueness of a distance-regular graph with intersection array {32,27,8,1;1,4,27,32} and related results

It is known that, up to isomorphism, there is a unique distance-regular graph $\Delta$ with intersection array {32,27;1,12} (equivalently, $\Delta$ is the unique strongly regular graph with parameters (105,32,4,12)). Here we investigate the distance-regular antipodal covers of $\Delta$. We show that, up to isomorphism, there is just one distance-regular antipodal triple cover of $\Delta$ (a graph $\hat\Delta$ discovered by the author over twenty years ago), proving that there is a unique distance-regular graph with intersection array {32,27,8,1;1,4,27,32}. In the process, we confirm an unpublished result of Steve Linton that there is no distance-regular antipodal double cover of $\Delta$, and so no distance-regular graph with intersection array {32,27,6,1;1,6,27,32}. We also show there is no distance-regular antipodal 4-cover of $\Delta$, and so no distance-regular graph with intersection array {32,27,9,1;1,3,27,32}, and that there is no distance-regular antipodal 6-cover of $\Delta$ that is a double cover of $\hat\Delta$.


Introduction
A graph Γ is distance-regular with intersection array {b 0 , b 1 , . . . , b d−1 ; c 1 , c 2 , . . . , c d } if Γ is connected with diameter d, and for i = 0, 1, . . . , d, whenever vertices v, w are at distance i in Γ, there are exactly b i vertices adjacent to w at distance i + 1 from v and exactly c i vertices adjacent to w at distance i − 1 from v (with the convention that b d = c 0 = 0). Distance-regular graphs occur in many areas of discrete mathematics, including coding theory, design theory and finite geometry. See, for example, the encyclopaedic reference [2] by Brouwer, Cohen and Neumaier. The present paper is a contribution to the problem of determining the distance-regular graphs with a given intersection array. We shall investigate the antipodal distance-regular covers of the Goethals-Seidel graph [11], the unique (up to isomorphism) distance-regular graph ∆ with intersection array {32, 27; 1, 12} [4,6]. The graph ∆ can be constructed as the second subconstituent of the second subconstituent of the famous McLaughlin graph, the unique distance-regular graph with intersection array {112, 81; 1, 56} (see [11] and [6]). A distance-regular antipodal triple cover∆ of ∆ was constructed by the author in [16], but its uniqueness was not determined. The main purpose of this paper is to prove that, up to isomorphism,∆ is the unique distanceregular graph with intersection array {32, 27, 8, 1; 1, 4, 27, 32}.
In the process of doing this, we confirm an unpublished result of Steve Linton which shows that there is no distance-regular graph with intersection array {32, 27, 6, 1; 1, 6, 27, 32}. We also prove there is no distance-regular graph with intersection array {32, 27, 9, 1; 1, 3, 27, 32}, and that there is no distanceregular antipodal 6-cover of ∆ that is a double cover of the distance-regular triple cover∆. We classify distance-regular antipodal r-covers of ∆ by studying the r-fold topological covers of the 2-dimensional simplicial complex whose 0-, 1-, and 2-simplices are respectively the vertices, edges, and triangles of ∆. Our main tool is version 2.0 of the author's GAP program described in [14] for the computation of fundamental groups, certain quotients of fundamental groups, and covers of finite abstract 2-dimensional simplicial complexes. This program is freely available from [18], where we also provide a GAP/GRAPE [7,17] logfile of all the computations described in this paper. It is hoped that [18] and the methods of this paper will be useful in further classifications of covers of graphs. All graphs in this paper are finite and undirected, with no loops and no multiple edges. Throughout, we follow [2] for graph-theoretical concepts and notation. An important new reference for distance-regular graphs, covering developments since [2] was published, is van Dam, Koolen and Tanaka [5]. See also Brouwer [1]. A good reference for the group-theoretical concepts used in this paper is Robinson [15]. We denote the commutator subgroup of a group G by [G, G], the cyclic group of order n is denoted by C n , and where p is a prime, the elementary abelian group of order p k is denoted by C k p .

The fundamental group and covers
Let Γ be a non-empty connected graph. We also consider Γ to be an abstract 2-dimensional simplicial complex (or 2-complex) whose 0-, 1-, and 2-simplices are respectively the vertices, edges, and triangles of Γ. With this in mind, thoughout this paper, by a cover of Γ, we mean a pair (Γ, θ), whereΓ is a graph and θ : V (Γ) → V (Γ) is a surjection, called a covering map, such that the following hold: • for every v ∈ V (Γ), the fibre vθ −1 of v is a coclique (independent set) ofΓ, • the union of any two distinct fibres mapping under θ to a non-edge of Γ is a coclique ofΓ, • the induced subgraph on any two fibres mapping under θ to an edge of Γ is a perfect matching inΓ, • the induced subgraph on any three fibres mapping under θ to a triangle of Γ consists of pairwise disjoint triangles inΓ.
Note that, as defined here,Γ is a cover of Γ precisely whenΓ is a topological cover of Γ when they are both viewed as 2-complexes as described above.
When we do not need to specify the covering map explicity, we may denote a cover (Γ, θ) simply byΓ. If each fibre of a coverΓ of Γ has the same positive integer cardinality r, then we callΓ an r-cover of Γ. A 2-cover is also called a double cover, and a 3-cover is also called a triple cover. If Γ is a non-complete graph, then an antipodal r-coverΓ of Γ is a connected r-cover of Γ with the property that being equal or at maximum distance inΓ is an equivalence relation on V (Γ), whose equivalence classes are the fibres. The connected r-covers of Γ correspond to the transitive permutation representations of degree r of the fundamental group G of Γ (regarded as a 2-complex), defined with respect to a fixed, but arbitrary, spanning tree of Γ. We shall explain this further in what follows. For a more general and detailed exposition, see [14], on which our explanation is based. We note that the choice of fixed spanning tree does not affect the isomorphism class of the fundamental group of Γ. Now, fix a spanning tree T of Γ. Then, for each arc (ordered edge) (v, w) of Γ, define an abstract group generator g v,w . Then G is the finitely presented group whose generators are these g v,w and whose relations are: • g s,t = 1 for each arc (s, t) of T ; • g v,w g w,v = 1 for each edge {v, w} of Γ; • g x,y g y,z g z,x = 1 for each triangle {x, y, z} of Γ.
Suppose (Γ, θ) is any connected r-cover of Γ. Then T θ −1 consists of r disjoint copies T 1 , . . . , T r of T , with T i θ = T . We may thus label the vertices ofΓ by ordered pairs (v, i), with v ∈ V (Γ) and i ∈ I = {1, . . . , r}, so that the In particular, note that a different choice for the fixed spanning tree T would only result in a relabelling of vertices within the fibres ofΓ. Now for each arc (v, w) of Γ, let ρ v,w be the permutation of I defined by z} is a triangle of Γ, then {x, y, z}θ −1 is a disjoint union of triangles inΓ, and so ρ x,y ρ y,z ρ z,x is the identity permutation. It now follows that the map ρ defined on the generators of G by (g v,w )ρ = ρ v,w extends to a transitive permutation representation from G to the symmetric group S r . The r-cover Γ is completely defined by this representation. Conversely, every transitive permutation representation ρ : G → S r defines a connected r-cover, denoted Γ ρ , of Γ. For each v ∈ V (Γ), the fibre of v in Γ ρ consists of the ordered pairs (v, i), for i ∈ I, and (v, i) and (w, j) are joined by an edge inΓ if and only if {v, w} is an edge of Γ and j = i((g v,w )ρ). The preimage in G of the stabilizer in Gρ of a point i ∈ I is isomorphic to the fundamental group of Γ ρ (see [14]). We consider two covers (Γ 1 , θ 1 ) and (Γ 2 , θ 2 ) of Γ to be isomorphic if there is a graph isomorphism from Γ 1 to Γ 2 which maps the fibres of Γ 1 to those of Γ 2 . Thus, isomorphic covers differ only by a relabelling of the fibres and of the vertices within each fibre. In particular, every connected r-cover of Γ is isomorphic to a cover of the form Γ ρ , for some transitive permutation representation ρ : G → S r , corresponding to the preimage in G of the stabilizer in Gρ of the point 1.
Isomorphism of covers can be checked as follows using nauty [13], which can be called from within GRAPE [17]. Given the cover Γ 1 of Γ, we make a {red, blue}-vertex-coloured graph Γ + 1 . The red-coloured vertices of Γ + 1 are the vertices of Γ 1 , with two red vertices joined in Γ + 1 precisely when those vertices are joined in Γ 1 . The blue-coloured vertices of Γ + 1 are in one-to-one correspondence with the fibres of Γ 1 , and a blue vertex is joined in Γ + 1 only to the red vertices in the corresponding fibre. Similarly, we make Γ + 2 from the cover Γ 2 of Γ. Then Γ 1 and Γ 2 are isomorphic as covers of Γ if and only if Γ + 1 is isomorphic to Γ + 2 by a colour-preserving graph isomorphism. We are usually interested in classifying covers up to isomorphism. First note that if two graphs are isomorphic, then every cover of the first is isomorphic to some cover of the second. Moreover, if (Γ 1 , θ 1 ) and (Γ 2 , θ 2 ) are isomorphic covers of Γ, and (Γ 1 ,θ 1 ) is any cover of Γ 1 , then (Γ 1 ,θ 1 θ 1 ) is a cover of Γ isomorphic to (Γ 2 ,θ 2 θ 2 ) for some cover (Γ 2 ,θ 2 ) of Γ 2 .

Imprimitivity and covers
We may sometimes be interested in a cover of a cover. Let Γ be a connected graph, let (Γ 1 , θ 1 ) be a connected m-cover of Γ, and let (Γ 2 , θ 2 ) be a connected n-cover of Γ 1 . Then clearly, (Γ 2 , θ 2 θ 1 ) is a connected mn-cover of Γ. Given a spanning tree T of Γ, we take a spanning tree T 1 of Γ 1 containing the forest T θ −1 1 , and define the fundamental group G of Γ with respect to T and the fundamental group G 1 of Γ 1 with respect to T 1 . Then (Γ 1 , θ 1 ) corresponds to a transitive representation ρ 1 : G → S m , with the fibre of v in Γ 1 being labelled {(v, i) : 1 ≤ i ≤ m}, as described previously, and (Γ 2 , θ 2 ) corresponds to a transitive representation ρ 2 : G 1 → S n , with the fibre of (v, i) being labelled {(v, i, j) : 1 ≤ j ≤ n}. Now (Γ 2 , θ 2 θ 1 ) corresponds to a transitive representation ρ : G → S mn , and if m, n > 1, then Gρ acts imprimitively on the indices (i, j) of the fibre of a vertex of Γ, the blocks of imprimitivity being {(i, j) : 1 ≤ j ≤ n}, for i = 1, . . . , m. Conversely, suppose that m, n > 1 and ρ : G → S mn is a transitive permutation representation of the fundamental group G of Γ such that Gρ is an imprimitive group having m blocks of size n. Then, if σ : G → S m is the transitive permutation action of Gρ on the blocks of imprimitivity, we see that Γ ρ is an n-cover of the m-cover Γ σ of Γ. We have the following: Theorem 3.1. Let Γ be a non-complete distance-regular graph and suppose Γ ρ is a distance-regular antipodal mn-cover of Γ corresponding to a transitive permutation representation ρ of the fundamental group G of Γ, such that Gρ has m > 1 blocks of imprimitivity of size n > 1. Then Γ ρ must be an n-cover of an antipodal distance-regular m-cover of Γ.

Proof. As above, we may suppose that
. Then each part in π is contained in a fibre of Γ ρ , and π is an equitable (also called regular) partition of V (Γ ρ ) (since each vertex in the part {(v, i, j) : 1 ≤ j ≤ n} is joined to exactly one or no vertex in the part {(w, k, ℓ) : 1 ≤ ℓ ≤ n}, with a join to one vertex precisely when {v, w} ∈ E(Γ) and ρ v,w maps the block B i to B k ). We may thus apply Theorem 6.2 of Godsil and Hensel [10] to deduce that Γ ρ /π is an antipodal distance-regular m-cover of Γ (see also Theorem 7.3 in [9, Section 11], attributed to Brouwer, Cohen and Neumaier).
To study covers of ∆, we explore quotients of the fundamental group of ∆, viewed as a 2-complex, with respect to a fixed spanning tree T of ∆. This fundamental group is denoted throughout this section by D. It was shown in [16] that D infinite. Proof. We compute a presentation for D/[D, D] using the program [18], and determine the abelian invariants of this group using GAP [7].
We are now in a position to classify the connected double covers of ∆, which was done independently by Steve Linton over 20 years ago, using his vector enumeration algorithm [12].
Theorem 4.2. Up to isomorphism of covers, ∆ has just 13 connected double covers, with each having its abelianised fundamental group isomorphic to C 15 2 × C 2 3 , and none being distance-regular.
Proof. Each subgroup of index 2 in D contains the commutator subgroup [D, D]. We compute the 2 16 − 1 covers of ∆ corresponding to the subgroups of index 2 in D/[D, D], and using GRAPE calling nauty, we determine that, up to isomorphism of covers, there are just 13 such covers. We test each of these covers for distance-regularity using GRAPE, and find that none of them are distance-regular. We use the program [18] to compute each of their abelianised fundamental groups.
We now consider the triple covers of ∆, and start by showing that D has no quotient isomorphic to the symmetric group S 3 . This is a corollary of the following:  We are now in a position to classify the connected triple covers of ∆.
Theorem 4.6. Up to isomorphism of covers, ∆ has just two connected triple covers, ∆ * and∆. The cover ∆ * has abelianised fundamental group isomorphic to C 16 2 × C 2 3 and is not distance-regular. The cover∆ has abelianised fundamental group isomorphic to C 18 2 × C 2 3 and is distance-regular.
Proof. Each connected triple cover of ∆ is isomorphic to a cover of the form ∆ ρ , for some transitive permutation representation ρ : D → S 3 . Since D has no quotient isomorphic to S 3 , we must have Dρ ∼ = C 3 , the cyclic group of order 3, and so each subgroup of index 3 in D is normal and contains [D, D]. We compute the four covers of ∆ corresponding to the subgroups of index 3 in D/[D, D], and determine that, up to isomorphism of covers, there are just two such covers, ∆ * and∆, and we calculate that these have the properties as stated in the theorem. Further properties of this distance-regular graph∆ are given in [16] (where the graph is called Λ). We now show there is no distance-regular antipodal 4-cover of ∆. Thus, we must have∆ isomorphic to∆. We computed all 2 18 − 1 2-covers of ∆ as covers of the form ∆ ρ of ∆, for ρ a permutation representation from D to S 6 . Just three of these covers ∆ ρ have Dρ isomorphic to A 4 , corresponding to the three permutation isomorphic representations of degree 6 of one quotient of D isomorphic to A 4 . Now given a connected 6-cover ∆ ρ with Dρ ∼ = A 4 , we construct the cover ∆ σ of ∆ defined by the representation of Dρ acting by right multiplication on the four (right) cosets of a Sylow 3-subgroup. Up to isomorphism of covers of ∆, there is only one such ∆ σ , which we find is not distance-regular.
Finally, we can prove the following: Theorem 4.9. There is no distance-regular antipodal 6-cover of ∆ that is a double cover of∆.
Proof. When we determined all 2 18 − 1 connected 2-covers of∆, we found that none is distance-regular.
This still leaves open the possibility of a distance-regular antipodal 6-cover of ∆ corresponding to a primitive degree 6 permutation representation of D.