Distance regularity in direct-product graphs

K e y w o r d s D i s t a n c e r e g u l a r graphs, Direct product. 1. I N T R O D U C T I O N E v e r y c o n n e c t e d g r a p h is r e p r e s e n t a b l e by m e a n s o f a level diagram (cf. [1]) as follows. C h o o s e a v e r t e x u, a n d le t i t be t h e sole r e s iden t of level zero. T h e ve r t i c e s on level i a re p rec i se ly t h o s e w h o s e d i s t a n c e f r o m u is i. N o w a d d edges of t h e g r a p h a n d n o t e t h a t t h e edges o c c u r o n l y b e t w e e n v e r t i c e s of a d j a c e n t levels and a m o n g ve r t i ce s of t h e s a m e level. Distance regularity is d e f i n a b l e in t e r m s of a level d i a g r a m . Le t d be t h e d i a m e t e r of a g iven g r a p h G, and le t v be a v e r t e x o f G on level i. F u r t h e r , le t ai = n u m b e r of ve r t i c e s on level i a d j a c e n t to v, i = 1 . . . . , d, bi. = n u m b e r of ve r t i ce s on level i + 1 a d j a c e n t to v, i = 0, . . , d 1, ci = n u m b e r of ve r t i c e s on level i 1 a d j a c e n t to v, i = 1, . . , d. We got to know about the unique characteristic of the (3, 12)-cage from C. Godsil and received much-needed encouragement from P. Weichsel. We are also thankful to the referee, whose comments on the earlier draft led to an improvement in the paper. *Author to whom all correspondence should be addressed (on leave, etc. from Delhi Institute of Technology, Delhi, India 0893-9659/1999/$ see front matter. @ 1999 Elsevier Science Ltd. All rights reserved. Typeset by A2t4S' I~ PII: S0893-9659(99) 00144-5