Geometric Antipodal Distance-Regular Graphs with a Given Smallest Eigenvalue

Abstract

In this paper we show that for a given integer $$m\ge 3$$, there are only finitely many antipodal distance-regular graphs with $$c_2\ge 2$$, odd diameter at least 5 and smallest eigenvalue at least $$-m$$. To prove this we first show that for a given integer $$m\ge 3$$, there are only finitely many antipodal distance-regular graphs with $$c_2\ge 2$$, diameter at least 5 and smallest eigenvalue at least $$-m$$ such that folded graphs are non-geometric. (A non-complete distance-regular graph is called geometric if it is the point graph of a partial linear space in which the set of lines is a set of Delsarte cliques.). Moreover, we show tight bounds for the diameter of geometric antipodal distance-regular graphs with an induced subgraph $$K_{2,1,1}$$ by studying parameters for geometric antipodal distance-regular graphs.