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A new condition for distance-regular graphs

✍ Scribed by J.H. Koolen


Publisher
Elsevier Science
Year
1992
Tongue
English
Weight
109 KB
Volume
13
Category
Article
ISSN
0195-6698

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Let F be a distance-regular graph with valency k (k >I 2) and diameter at least 2, and denote by ;t 1 and 2%~m the second largest and least eigenvalue of F, respectively. Assume the multiplicity m( )O of some eigenvalue ;~ ( )~ :/: k) of F satisfies m( Z ) < k. Then ;~ = Z 1 or )'rot. and either (i)

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Let ⌫ be a distance-regular graph with a 1 ΟΎ 0 , r Ο­ max Ν• j 3 ( c j , a j , b j ) Ο­ ( c 1 , a 1 , b 1 ) Ν– Ρƒ 2 and a i Ο­ a 1 c i , for 1 Ρ€ i Ρ€ 2 r . Take any u and in ⌫ at distance r Ο© 1 . We show that there exists a collinearity graph of a generalized 2( r Ο© 1)-gon of order ( a 1 Ο© 1 , c r Ο© 1 Οͺ 1)