GEOMETRIC DISTANCE-REGULAR COVERS

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1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 22 (1993), GEOMETRIC DISTANCE-REGULAR COVERS C.D. G o d s i l 1 (Received March 1993) Abstract. Let G be a distance-regular graph with valency k and least eigenvalue r. Delsarte proved that a clique in G has cardinality at most 1 ~. W e call a distanceregular graph geometric if each edge lies in a unique clique of this cardinality. Any geometric distance-regular graph is the point graph of a partial linear space with the property that the number of points on a line closest to a given point only depends on the distance between the point and the line. W e derive some conditions which imply that a distance-regular graph is geometric, and use them to show that the number of antipodal distance-regular graphs with diameter three and given least eigenvalue is finite. 1. Introduction The work reported here arises from a study of the antipodal distance-regular graphs of diameter three. Such graphs are covering graphs of complete graphs and are interesting in part because of their connection to problems in finite geometry and design theory. These graphs are discussed in Sections 12.5 and 12.7 of [2], and at some length in [5]. Our results are motivated by the characterisation of strongly regular graphs presented by Neumaier in [9]. Briefly, he shows that with a finite number of exceptions a strongly regular graph with given least eigenvalue must be derived from either a Steiner system or an orthogonal array. Neumaier s results are based on classical work of Bose [1]. Starting from [1], and a recent interesting extension of it due to Metsch [8], we sketch a theory of geometric distance-regular graphs in Section 2 of this paper. In Section 3 we use this to show that there are only finitely many antipodal distanceregular graphs with given least eigenvalue. We also show how it can be used to show particular parameter sets cannot be realised. 2. Claws and Cliques Let G be a distance-regular graph with valency k and diameter d and let..., 6d denote the eigenvalues of G, in nonincreasing order. It follows from work of Delsarte that a clique in G has cardinality at most and if equality holds then it is a completely regular subset of G and no vertex of G is at distance d from C. (If u and v are vertices at distance d in G, let Cd denote the number of vertices adjacent to v and at distance d 1 from u. For a vertex at distance d 1 from C there will be exactly Cd/Od vertices in C at distance 1991 A M S Mathematics Subject Classification: Primary 05E30, Secondary 05C50. 1 Support from grant OGP of the National Sciences and Engineering Council of Canada is gratefully acknowledged.

2 32 C.D. GODSIL d 1 from it.) A clique in a distance-regular graph with cardinality given by (1) will be called a Delsarte clique. An s-claw in a graph G is an induced subgraph isomorphic to K\,s. Any s-claw is determined by an independent set of size s in the neighbourhood of some vertex. We say an s-claw is maximal if it is not contained in an (s + l)-claw. An incidence structure of points and lines is a 1 -design if there are integers s and t such that each point is on exactly t + 1 lines and each line is incident with exactly s + 1 points, and is a partial linear space if all lines contain at least two points and any two points lie in at most one line. Suppose V is a partial linear space which is also a 1-design, with parameters s and t. Then the point graph G of V is regular with valency s(t + 1). If B is the incidence matrix of the partial linear space then the adjacency matrix A(G) of G is B B T ~ { t + 1)1. Therefore the least eigenvalue of A = A(G) is at least t 1 and equality holds if and only if B B T is not invertible, i.e., its rows are linearly dependent. If s > t then B has more rows than columns and so the least eigenvalue is always t 1 in this case. The lines of V determine a set of cliques in G with cardinality s + 1 such that each vertex lies in exactly t + 1 of these cliques, and two of the cliques have at most one vertex in common. If the point graph of G is distance-regular and t 1 is its least eigenvalue then these cliques are Delsarte cliques. The partial linear space V then has the property that for a point p and a line, the number of points on nearest to p only depends on the distance between p and. (If this number is always one then G is a near polygon. A necessary and sufficient condition for this is that Qd ^d or equivalently that Cd = t + l.) We will call a distance-regular graph geometric if it is the point graph of a partial linear space and the cliques determined by the lines are Delsarte cliques. (Such a partial linear space must be a 1-design.) The Johnson graph J(v,k) provides a simple example of a geometric distance-regular graph. It may be viewed as the point graph of an incidence structure with the k-subsets of { 1,..., *;} as its points and the (k l)-subsets as its lines, with a (k l)-subset incident with all fc-subsets which contain it. A graph is amply regular if there are constants, ai and C2 say, such that any two adjacent vertices have exactly a\ common neighbours and any two vertices at distance two have exactly C2 common neighbours. Any distance regular graph is amply regular. Our first result is a local condition which can be used to show that an amply regular graph is the point graph of a partial linear space (c/. Metsch [8] and Bose [1].) Lem m a 2.1. Let G be an amply regular graph and let N be the neighbourhood of a vertex in G. Suppose that the maximum number of vertices in an independent set in N is s. If a i + 1 (2s - 1 ) ( c2-1 ) > 0 then N can be partitioned into s vertex disjoint cliques, and every vertex in N lies in an independent set of size s.

3 GEOMETRIC DISTANCE-REGULAR COVERS 33 P roof. Observe that any two non-adjacent vertices in the N have at most c2 1 common neighbours in N. Let S be an independent subset of the neighbourhood N, with cardinality s, and let x be a vertex in S. Then there are at least ai (s 1) (c2-1) > s(c2-1) vertices in N adjacent to x and to no other vertex in S. If y and z are two of these vertices they are adjacent - otherwise (,S \ x ) U { y,z } is an independent set in N with cardinality s + 1. Thus N contains cliques of size at least 1 + ai (s l)(c2 1). We call such cliques big. Note that each vertex of S lies in a big clique. Any two maximal big cliques in N are vertex disjoint. For suppose C\ and C2 are two distinct maximal cliques in N such that C\ ft C2 ^ 0. Then \C\ U C2\< ai +1 and Ci n C 2 < c2 1, whence This is impossible if Ci and C2 are big. \C\ + C21< ai + C2- Any maximal clique C intersects every maximal independent set in N. For let T be a maximal independent set. If some vertex in C is not adjacent to any vertex in T then T can be extended to an independent set of size s + 1. If some vertex in T is adjacent to all vertices in C then C is not a maximal clique. It follows that each vertex of T has at most C2 1 neighbours in C and, since T < s, that \C\ < s{c2-1). Once again this constraint cannot be satisfied by a big clique. We now complete the proof. There are s pairwise disjoint big cliques which cover the vertices of S. If v is a vertex of N then it lies in a maximal independent set, T say, of size at most s. Since each of the s big cliques meeting S must intersect T, we see that \T\ s and T is contained in the union of these cliques. C orollary. Let G be a distance-regular graph with least eigenvalue r. If there are no s-claws in G with s > r and ai + 1 (2 r - l)(c2-1) > 0 then G is geometric. P roof. Denote r by t. Let k be the valency of C, let u be a vertex in G and let S be a maximal independent set in the neighbourhood N oi u with cardinality s. By Delsarte s bound, any clique in C has cardinality at most

4 34 C.D. GODSIL and so any clique in N has cardinality at most k/t. From the lemma we see that N is covered by s cliques, whence s k /t > k and s > t. Thus the maximum size of an independent set in the neighbourhood of a vertex in G is t, and there is a set of Delsarte cliques in G such that every vertex lies in k /t of them, and each edge of G lies in exactly one. Hence G is the point graph of the incidence structure formed by the vertices of G, together with the above set of Delsarte cliques as lines. The next lemma provides an upper bound on the independence number of the neighbourhood of a vertex in an amply regular graph. Lemma 2.3. Let G be an amply regular graph. If G contains an s-claw then k > s(ai + 1) - Q ( C2 ~ 1)> Proof. Let N be the neighbourhood of the vertex u in G and suppose that S = {vi,..., vs} is an independent set in N. Let Mi denote the set of neighbours of u adjacent to vt. By standard sieve inequalities (see, e.g., [7; Exercise 2.9]) S S and consequently M t\+ \M j n M k\ i = 1 i=l l<j<k<s sai < k s + Q (C2 ~!) There is an alternative bound of the same general form, which is sometimes stronger. The following argument comes from Metsch [8]. Let G be an amply regular graph such that the maximum size of a claw is s and let N be the neighbourhood of a vertex x in G. Assume a\ + 1 (2s l)(c2 1) > 0. Then the vertex set of N is covered by s vertex disjoint big cliques, each with cardinality at least a\ + 1 (s l)(c2 1). Let C be one of these cliques, let u be a vertex in it and let v be a vertex adjacent to u and not in G. If w is a vertex in C \ u adjacent to v then {i>,k;} is contained in a big clique, and the cliques obtained as w varies over the neighbours of v in C intersect pairwise in v. Hence v has at most s neighbours in C. Now consider the s big cliques in the neighbourhood of x. There are s 1 which do not contain v, and v has at most s 1 neighbours in each of these. Let Cv be the unique big clique on u containing v. Then the valency of v in the neighbourhood of u is at most \CV\ 1 + (s l)2, i.e., I fli \CV\ 1 + (s l)2. Summing this inequality over the s big cliques on u then yields Metsch s bound: s(ai + 1) < k + s(s l)2. Finally we mention another result which can sometimes be employed to show that a distance-regular graph is geometric.

5 GEOMETRIC DISTANCE-REGULAR COVERS 35 Lemma 2.3. (Brouwer and Neumaier [3]). Let G be an amply regular graph such that C2 = 2. If k < ai(ai + 3)/2 then G contains no copies of -# 2,1,1- I It is easy to show that the neighbourhood of a vertex in a distance-regular graph which contains no copies of # 2,1,1 is a disjoint union of cliques of size ai + 1. Note also that a distance-regular graph with C2 = 1 contains no -# 2,1,1- However the absence of a copy of -# 2,1,1 does not guarantee that the graph is geometric - for this we need that a clique of size ai + 2 is a Delsarte clique, i.e., k (ai + l) 0d - 3. Covers of Complete Graphs We now apply the machinery of the first section to the study of antipodal distance-regular graphs with diameter three. Any such graph is a covering graph of a complete graph, and we will often refer to them simply as covers. For the basic theory, see [5] and [2; Sections 1.10,12.5 and 12.7]. These graphs can be parameterised by triples (n, r, C2), where n is the number of vertices in the underlying complete graph, r is the index of the cover and C2 has its usual meaning, i.e., the number of common neighbours of two vertices at distance two. The valency of a cover is n 1 and the number of common neighbours of two adjacent vertices is ai, which is determined by the identity n 2 ai = (r l)c2. A cover of K n is distance-regular with diameter three, and so has exactly four distinct eigenvalues. Two of these are n 1 and 1, the remaining two will be denoted by 9 and r. They are the zeros of the quadratic from which we may deduce that x 2 - (ai - c2)x - (n - 1), n 1 = 9t, rc2 = (9 + l)(r + 1). The first of these identities show that 9 and r have opposite signs. We will always assume that 9 > 0. The multiplicity of 9 as an eigenvalue of G is (r 1 )nr t 9 If a\ = C2 then 9 r, whence 9 and r are the two square roots of n 1. The Delsarte bound for cliques in G is The index r of a cover of K n cannot be greater than n 1. r Lemma 3.1. Let G be an antipodal distance-regular graph of diameter three. If 9 > max (C2 + I), 2 c 2 ( 1 ),

6 36 C.D. GODSIL where t r, then G is geometric. P roof. As a\ = c2 4-9 t the inequality 9 > 2 c2(i 1) t implies that 0 < ai c2 + t 2c2( 1) + 1 = a\ + c2 2t(c2 1). Therefore, if we can show that there are no (t + l)-claws in G, it follows from Corollary 2.2 that G is geometric. Suppose by way of contradiction that G contains an independent set of size t + 1 in the neighbourhood N of some vertex. By Lemma 2.3 we then have (t ) (c2 1) < 71 1 = t9. Subtituting a\ = c2 + 9 t and rearranging, we obtain that 9 < - ( t + 1)( - 2 ) ( c2 + 1). This contradicts our hypothesis on 9. We next describe an important construction due to Brouwer. (See [2 ; Proposition ].) Let H be the point graph of a generalised quadrangle G Q (s,t). A spread in H is a set of cliques of size s + 1 which partition the vertices of H. (Any spread consists of exactly st + 1 cliques.) If G is obtained from H by deleting the edges from each clique in the spread, then G is an antipodal distance-regular graph of diameter three, with parameters (s +l,s + l, 1). More generally this construction works if H is a strongly regular graph with the same parameters as the point graph of a generalised quadrangle, and any cover with the parameter set just given arises in this way. It is also not difficult to show these covers can be characterised by the condition that r = Lemma. Let G be an antipodal distance-regular graph of diameter three. If G is geometric then r divides 9 + 1, and if r = then G arises by deleting the edges in a spread from the point graph of a generalised quadrangle. P roof. If C is a Delsarte clique in a distance-regular graph, any vertex u not in C but adjacent to a vertex of C has exactly ic, + ( ^ T + 1 neighbours in C. (See, e.g. [2; Proposition 4.4.6(i)].) As rc2 = (0 + 1)(t + 1) and \C\ 9 + 1, this means that u has exactly [9-1-1 )/r neighbours in C. This proves our first claim; the second follows from our remarks above. Geometric covers with r a proper divisor of are constructed from generalised quadrangles with spreads in [6].

7 GEOMETRIC DISTANCE-REGULAR COVERS 37 Theorem 3.3. If r < 1 then there are only finitely many antipodal distanceregular graphs with diameter three and least eigenvalue r. P roof. Suppose G is an antipodal distance-regular graph with parameters (n, r, C2). As r must be less than n and n 1 = 9t, the lemma can be proved by showing that 9 is bounded above by a function of r. If r is not an integer then r and 9 must have equal multiplicity, from which it follows that a\ = c2 and n 1 = t 2. Thus we may assume that r and 9 are integers. We will write t for r. The multiplicity of 9 as an eigenvalue of G is n (r-l)t = { r _ 1)t2 _ ( r - W 3 ~ t ) 9 + t v ' 9 + t v If r = 2 this implies that 9 + t divides t3 t, whence 9 < t3 21. Thus we may assume that r > 3. From (1) we find that 9 + t divides (r l)(t3 t). If me < n+i 3 then, by [5; Lemma 3.5], must divide c2. Since rc2 = (9 + l)(t 1) this implies that r divides t 1, whence (1) yields that 9 < t4 t2 t. Thus we may assume that me > n + r 3. Then me > n and using (1) again we deduce that (r l)t/(9 + t) > 1. Consequently r > + 2 and hence we deduce that c2 < t2 t. Applying Lemma 3.1 we now infer that if 9 is not bounded above by a function of order at most t4 then G is geometric. We now show that if G is geometric then 9 is still bounded by a function of t. Since G is geometric r divides 9 + 1, by Lemma 3.2. Suppose that r (9 + l)/s. We may assume that 9 > t4, which implies that me > n + r 3 and r > f -f 2. This also yields that s < t. From (1) we have m e - ( r - \ ) t As me is an integer, this implies 9 + t is no greater than (t + s 1 )(t3 t). Since s < t we find that 9 < (2t 2 )(t3 t). I We have no compelling reason to believe that a geometric cover can have 9 greater than t2 + t. The machinery we have developed can also be used to show that particular parameter sets are not feasible. By way of example, consider the parameter set (28,10,2), which satisfies all the constraints given in [5; Section 3]. Since c2 = 2, we can apply Lemma 2.4 to deduce that a cover G with these parameters contains no copy of i f 2,1,1, and hence the neighbourhood of any vertex in G is a union of three vertex disjoint cliques of cardinality a\ + 1 = 9. Since 9 = 9, this means that each vertex lies in exactly three Delsarte cliques, and so G is geometric. Since no G Q (s2,s) contains a spread (see e.g., [10; 1.8.3]) we deduce that there is no cover with the given parameters. A similar argument disposes of graphs with parameter set (19, 7,2) - here a\ = 5 and 9 = 6 and so such a graph is geometric by Lemma 2.4. Since r = this means that the graph comes from a G Q (6,3) with a spread. By Dixmier and Zara [4] (or [10; Section 6.2]), no G Q (6,3) exists. Lemma 3.1 shows that a cover with parameters (105,27,3) is geometric. Since such a cover has 9 = 26, it therefore must come from a G Q (26,4), which does not

8 38 C.D. GODSIL exist. Similarly a cover with parameters (141,27,4) is geometric by Lemma 3.1, and therefore cannot exist since r does not divide = 36. A cknow ledgem ent. I would like to thank Tilla Schade for her careful reading of an earlier version of this work. This has spared me some embarrassment, and the reader some confusion. References [1] R.C. Bose, Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math. 13 (1963), [2] A.E. Brouwer, A.M. Cohen and A. Neumaier, Distance-Regular Graphs, Springer, Berlin, [3] A.E. Brouwer and A. Neumaier, A remark on partial linear spaces of girth 5 with an application to strongly regular graphs, Combinatorica 8 (1988), [4] S. Dixmier and F. Zara, Etude d un quadrangle generalise autour de deux de ses points non lies, preprint (1976). [5] C.D. Godsil and A.D. Hensel, Distance regular covers of the complete graph, J. Combinatorial Theory B 56 (1992), [6] C.D. Godsil, Krein covers of complete graphs, Australasian J. Comb. 6 (1992), [7] L. Lovasz, Combinatorial Problems and Exercises, North-Holland, Amsterdam, [8] Klaus Metsch, Improvement of Bruck s completion theorem, Designs, Codes and Cryptography 1 (1991), [9] A. Neumaier, Strongly regular graphs with least eigenvalue ra, Archiv der Mathem. 33 (1979), [10] S.E. Payne and J.A. Thas, Finite Generalized Quadrangles, Pitman, Boston, C.D. Godsil Department of Combinatorics and Optimization University of Waterloo Waterloo Ontario C A N A D A N 2 L 3G 1 chris@dibbler.uwaterloo.ca