An Eigenvalue Characterization of Antipodal Distance-Regular Graphs

Abstract

Let $G$ be a regular (connected) graph with $n$ vertices and $d+1$ distinct eigenvalues. As a main result, it is shown that $G$ is an $r$-antipodal distance-regular graph if and only if the distance graph $G_d$ is constituted by disjoint copies of the complete graph $K_r$, with $r$ satisfying an expression in terms of $n$ and the distinct eigenvalues.