Krein covers of complete graphs

Let G be an antipodal distance regular cover of a complete graph with index T'. If T' = 2 then it is known that the neighbourhood of a vertex in G is strongly regular, and the Krein bound is tight for G. We use the theory of spherical designs ( due to Delsarte, Goethals and Seidel) to show that if r > 2 and the Krein bound is tight for G then the neighbourhood of any vertex in G is strongly regular. Further, if e denotes the second largest eigenvalue of G then T' must divide () + 1, and if r = () + 1 then G arises by a standard construction from a generalised quadrangle with a spread. If e is a prime power and r divides e + 1, covers of index r for which the Krein bound holds can be constructed and in many cases the neighbourhoods of a vertex in these covers are new strongly regular graphs. 1. ANTIPODAL DISTANCE REGULAR COVERS We are going to study a class of distance regular covers of complete graphs. Thus we begin by explaining what a cover is, and what it means for one to be distance regular. (However we assume that the reader knows what a distance-regular graph is. A brief introduction will be found in [1], and an extensive treatment in [2]. Unsupported assertions about properties of antipodal distance regular covers of I{ n are taken from [8].) We say that a graph H is cover of a graph G if there is a surjection f from V (H) onto V (G) such that f maps edges of H to edges of G, and the restriction of f to the neighbourhood of any vertex is an isomorphism. (If we view G and H -:~ Support from grant OGP0093041 of the National Sciences and Engineering Council of Canada is gratefully acknowledged. Australasian Journal of Combinatorics §.( 1992) . pp.245-255 as topological spaces, this is equivalent to the topologist's definition of a cover, if it helps. ) If v E V(G) then j-l(v) called a fibre of the cover. When G is connected it is not hard to show that all fibres must have the same cardinality, which we call the index of the cover. In general the cover is not determined just by the graphs Hand G there may be more than one possible local isomorphism j from H to G, and so it may be necessary to explicitly identify the fibres. A cover H of G with diameter d is antipodal if two vertices of H are at distance d if and only they are in the same fibre. Informally, any cover of G wi th index r can be constructed as follows. For each vertex in G there is a corresponding set of r disjoint vertices in H, and if u and v are adjacent in G then the corresponding r-sets are joined by the edges of an r-matching. The cube is an antipodal cover of ]{4 with index two and the line graph of Petersen's graph is an antipodal cover of ]{s with index three. Both these covers have the property that the number of common neighbours of two vertices is determined by the distance between them. Any antipodal cover of ]{n with this property is distance regular, and must have diameter equal to three. If H is an antipodal distance regular cover of ]{ n then we denote the number of common neighbours of two adjacent points by aI' and the number of common neighbours of two points at distance two by c2 • The parameters n, r, a 1 and C2 are related by n 2 a 1 = (r 1)c2' (1) Thus a1 is determined by the other three parameters. An expression such as "H is a cover with parameters (n, r, c2 )" will indicate that H is an distance regular antipodal cover of ]{n with index r, and a 1 determined by (1). We note that any distance regular cover of ]{n with diameter three must be antipodal. Let H be an (n,r,c2) cover. Since 1I is distance regular with diameter three, its adjacency matrix A( H) has exactly four distinct eigenvalues. Since it is a cover of Knl two of these eigenvalues are n 1 and -1, with multiplicities 1 and n 1 respectively. The remaining two eigenvalues will be denoted by a and T, and are the zeros of the quadratic