Tight Distance-Regular Graphs

We consider a distance regular graph with diameter d and eigenvalues k d We show the intersection numbers a b satisfy k a d k a ka b a We say is tight whenever is not bipartite and equality holds above We charac terize the tight property in a number of ways For example we show is tight if and only if the intersection numbers are given by certain rational expressions involving d independent parameters We show is tight if and only if a ad and is homogeneous in the sense of Nomura We show is tight if and only if each local graph is connected strongly regular with nontrivial eigenvalues b and b d Three in nite families and nine sporadic examples of tight distance regular graphs are given CONTENTS Introduction Preliminaries Edges that are tight with respect to an eigenvalue Tight edges and combinatorial regularity The tightness of an edge Tight graphs and the Fundamental Bound Two characterizations of tight graphs The auxiliary parameter Feasibility A parametrization The homogeneous property The local graph Examples of tight distance regular graphs References Introduction Let X R denote a distance regular graph with diameter d and eigenvalues k d see Section for de nitions We show the intersection numbers a b satisfy k a d k a ka b a We de ne to be tight whenever is not bipartite and equality holds in We characterize the tight condition in the following ways Our rst characterization is linear algebraic For all vertices x X let x denote the vector in R with a in coordinate x and in all other coordinates Suppose for the moment that a let x y denote adjacent vertices in X and write w P z where the sum is over all vertices z X adjacent to both x and y Let denote one of d and let E denote the corresponding primitive idempotent of the Bose Mesner algebra We say the edge xy is tight with respect to whenever E x E y Ew are linearly dependent We show that if xy is tight with respect to then is one of d Moreover we show the following are equivalent i is tight ii a and all edges of are tight with respect to both d iii a and there exists an edge of which is tight with respect to both d Our second characterization of the tight condition involves the intersection numbers We show is tight if and only if the intersection numbers are given by certain rational expressions involving d independent variables Our third characterization of the tight condition involves the concept of homogeneous that appears in the work of Nomura See also Curtin We show the following are equivalent i is tight ii a ad and is homogeneous iii a ad and is homogeneous with respect to at least one edge Our fourth characterization of the tight condition involves the local structure and is reminiscent of some results by Cameron Goethals and Seidel and Dickie and Terwilliger For all x X let x denote the vertex subgraph of induced on the vertices in X adjacent x For notational convenience de ne b b d and b b We show the following are equivalent i is tight ii for all x X x is connected strongly regular with nontrivial eigenvalues b b iii there exists x X such that x is connected strongly regular with nontrivial eigenvalues b b We present three in nite families and nine sporadic examples of tight distance regular graphs These are the Johnson graphs J d d the halved cubes H d the Taylor graphs four fold antipodal covers of diameter constructed from Fisher groups p two fold antipodal covers of diameter constructed by Soicher a fold and a fold antipodal cover of diameter constructed by Meixner and the Patterson graph Thm which is primitive distance transitive and of diameter Preliminaries In this section we review some de nitions and basic concepts See the books of Bannai and Ito or Brouwer Cohen and Neumaier for more background information Let X R denote a nite undirected connected graph without loops or multiple edges with vertex set X edge set R path length distance function and diameter d maxf x y j x y Xg For all x X and for all integers i we set i x fy X j x y ig We abbreviate x x By the valency of a vertex x X we mean the cardinality of x Let k denote a nonnegative integer Then is said to be regular with valency k whenever each vertex in X has valency k is said to be distance regular whenever for all integers h i j h i j d and for all x y X with x y h the number phij j i x j y j is independent of x and y The constants phij are known as the intersection numbers of For notational convenience set ci p i i d ai p i i i d bi p i i d ki p ii i d and de ne c bd We note a and c From now on X R will denote a distance regular graph of diameter d Observe is regular with valency k k b and that k ci ai bi i d We now recall the Bose Mesner algebra Let MatX R denote the R algebra consisting of all matrices with entries in R whose rows and columns are indexed by X For each integer i i d let Ai denote the matrix in MatX R with x y entry Ai xy if x y i if x y i x y X Ai is known as the ith distance matrix of Observe A I A A Ad J J all s matrix Ati Ai i d AiAj d X