Eigenvalue multiplicities of highly symmetric graphs

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

$\text{k(k?1)}{}^{\mathrm{<mtext></mtext><mtext>g</mtext><mtext>4</mtext><mtext>]?1</mtext>}}$

if g≡0 or 1 (mod 4),

$\text{2(k?1)}{}^{\mathrm{<mtext></mtext><mtext>g</mtext><mtext>4</mtext><mtext>]</mtext>}}$

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, $\text{1?s?}\text{1}\text{2}\text{g]}$, then the multiplicity of any eigenvalue λ (λ≠±?k) is at least

$\text{k(k?1)}{}^{\mathrm{<mtext>s</mtext><mtext>2?1</mtext>}}$

if s is even,

$\text{2(k?1)}{}^{\mathrm{<mtext>(</mtext><mtext>s?1)</mtext><mtext>2</mtext>}}$

if s is odd.