ON PRIMITIVE GRAPHS AND OPTIMAL VERTEX ASSIGNMENTS

The r e s u l t s presented her: a r e an outgro;th of an i n v e s t i g a t i o n on t h e sub iec t of cube-n;mbering and some of i t s gene ra l i za t ions . Cube-numbering r e f e r s t o the gene r a l problem of a s s ign ing a given se t of r e a l va lues t o t h e v e r t i c e s of an n-cube ( o r more gene ra l ly , some graph G ) s o t h a t t h e sum over a l l edges e of t h e n-cube ( o r G ) of a given func t ion of the d i f f e r e n c e of t he va lues of t h e two endpoints of e i s minimized. This concept a r i s e s i n t h e s tudy of c e r t a i n optimal b inary codes ( c f . [41, [51, [71). I n t h i s no te we extend t h e s e t of admiss ib le graphs G (of which t h e n-cube i s now a s p e c i a l ca se ) and express t h e preceding ques t ion of optimal ve r t ex assignments i n numbert h e o r e t i c terms. The s o l u t i o n t o t h e corresponding number theory problem i s obtained by s tudying t h e s t r u c t u r e of a s p e c i a l c l a s s of graphs we c a l l p r i m i t i v e . Although t h e few f a c t s e s t ab l i shed here about p r i m i t i v e graphs a r e su f f i c i e n t t o completely answer our ve r t ex assignment problem, i t w i l l be seen t h a t many i n t e r e s t i n g open ques t ions remain.