Eigenvalue multiplicities of highly symmetric graphs

Abstract

We find lower bounds on eigenvalue multiplicities for highly symmetric graphs. In particular we prove: Theorem 1. If Γ is distance-regular with valency k and girth g (g≥4), and λ (λ≠±-k) is an eigenvalue of Γ, then the multiplicity of λ is at least k(k-1)[ g 4]-1 if g≡0 or 1 (mod 4), 2(k-1)[ g 4] if g≡2 or 3 (mod 4) where [ ] denotes integer part. Theorem 2. If the automorphism group of a regular graph Γ with girth g (g≥4) and valency k acts transitively on s-arcs for some s, 1≤s≤[ 1 2g], then the multiplicity of any eigenvalue λ (λ≠±-k) is at least k(k-1)s 2-1 if s is even, 2(k-1)( s-1) 2 if s is odd. © 1982.