Antipodal graphs of diameter three

Distance-regular graphs of diameter three are of three (almost distinct) kinds: primitive, bipartite, and antipodal. An antipodal graph of diameter three is just an r-fold covering of a complete graph K_k+1 for some r?k. Its intersection array and spectrum are determined by the parameters r, k together with the number c of 2-arcs joining any two vertices at distance two. Most such graphs have girth three. In this note we consider antipodal distance-regular graphs of diameter three and girth ? 4. If r=2, then the only graphs are “K_{k+1, k+1} minus a 1-factor.” We therefore assume r?3. The graphs with c=1 necessarily have r=k and were classified in lsqb3rsqb. We prove the inequality $\text{r?2>c}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ (Theorem 2), list the feasible parameter sets when c=2 or 3 (Corollary 1), and conclude that every 3-fold or 4-fold antipodal covering of a complete graph has girth three (Corollary 2).