Eigenvalue multiplicities of highly symmetric graphs

✍ Scribed by Paul Terwilliger

• Publisher: Elsevier Science
• Year: 1982
• Tongue: English
• Weight: 905 KB
• Volume: 41
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(82)90025-5

✦ Synopsis

We find we prove: lower kunds on eigenvalue multiplicities for highly symmetric graphs. In partictiar

I.. If r is distance-regular with valency k and girth g (g 2 4). and A (A # *k) IS an eigenvalue of r, then the multiplicity of h is at least k(& -

#e/41-1 if g=O or 1 ,'mod 4), 2( k -1)["4' ifs=2 or 3(mod4) where [ J denotes integer part. emem 2. If the automorphism group of a regular graph r with girth g (g 3 4) and valency k acts transitively on s-arcs for some s, 1 c s ~[$g], !hen the multiplicity of any eigenvalue h (A # *k) is a; least k(k -l)"'*-' if s is ewn, 2(& -l)(s -lb/2 ifs is odri.