By Ladislav NebeskyÌ
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Additional resources for Algebraic properties of trees
2. rx + sy Embed Cay(x,y : G ) Cay(x,y : G ) of in the torus. 3. knot(C) E Z Cay(x,y : G ) torus: Lift C in Cay(x,y : G) , we let C considered as an oriented knot on the to some path in the plane. of this path, and let in the natural way. For any elementary circuit C Z be the knot class of x on the torus by identifying a vertex with the point Let (a,b) be the initial endpoint (c,d) be the terminal endpoint. Then knot(C) = ( c - a , d-b). Intuitively, if times knot(C) = (m,n) , then m is the (algebraic) number of C wraps around the torus longitudinally, and n is the number of times C wraps around the torus meridionally [9, pp.
Murty, (Academic P r e s s , New York, 1 9 7 9 ) , 341-355. , 8 (19731, 367-387. V. D. A. I n t h i s p a p e r i t i s shown t h a t e v e r y c o n n e c t e d metac i r c u l a n t g r a p h h a v i n g a n e v e n number o f b l o c k s o f prime c a r d i n a l i t y , o t h e r t h a n t h e sole e x c e p t i o n o f t h e P e t e r s e n graph, h a s a Hamilton cycle. T h i s p a p e r i s a s e q u e l t o [31 in which i t was shown t h a t e v e r y c o n n e c t e d m e t a c i r c u l a n t g r a p h w i t h a n odd number o f v e r t i c e s g r e a t e r t h a n o n e a n d w i t h For purposes of b r e v i t y , we b l o c k s of p r i m e c a r d i n a l i t y h a s a H a m i l t o n c y c l e .
THEOREM 7 . 1 . 9(b), and o n l y i f , f o r some t , t h e digraph 0 to h a s knot class Ht(d) Theorem 7 . 1 i s a s p e c i a l case of t h e f o l l o w i n g r e s u l t . Hamilton p a t h i f and o n l y i f i t s knot c l a s s i s (0,O) - d(y-x) (0,O) Because . x if So is a Ht(d) (Lemma 5 . 8), t h i s r e s u l t a l s o g i v e s a Cay(x,y : G ) c h a r a c t e r i z a t i o n of t h e Hamilton p a t h s i n The knot class o f THEOREM 7 . 2 . and with d 0 5 t c IG :c y - x i l . H (d) i s equal t o Bt(d) t 0 5 d c ord(y-x) .
Algebraic properties of trees by Ladislav NebeskyÌ