Graph Theory in Modern Engineering: Computer Aided Design, by Henley

By Henley

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2 2"1 1 ' m -1 correspondins to r , r e t u r n s u s t o cl. corresponding t o in q cl+w-v ( i i ) Suppose t h a t e d g e s c o l o r e d C1 to Ci and Then t h e r e e x i s t s that 3 , 9 E Cl I t now follows t h a t , f o r is obtained. = hr . ) I n g e n e r a l , given a Cayley c o l o r graph i? I t h e v e r t i c e s a r e t h e r i g h t cosets o f coset g r a p h a s f o l l o w s : -_I r , so U. e. the right cosets) are the T/i2 the defining relations f o r ... ) hl qhl such t h a t and r e s t o r i n g t h e e d g e s c o l o r e d done unambiguously, hl (Ariain a s s u m e By s h r i n k i n g t h e s e c o m p o n e n t s , e a c h t o a (G).

4 c a l l e d weakly connected i f t h e ( u n d i r e c t e d ) pseudograph underlying i s connected. D F o r e x a m p l e , see F i g u r e 4-2, D' where D is strongly connected, -- i s u n i l a t e r a l l y connected ( b u t n o t s t r o n g l y c o n n e c t e d ) and D" i s weakly connected ( b u t n o t u n i l a t e r a l l y c o n n e c t e d ) . D: A D': D": F i g u r e 4-2. 4-2. Autornorphisrns W e h a v e p r e v i o u s l y d e f i n e d a n automorphisrn of a g r a p h V(G) p e r m u t a t i o n of preserving adjacency).

See Figure 5-3 for the sphere, open cylinder, torus, projective plane, mobius strip, and klein bottle, respectively. The top three 2-manifolds are orientable, the bottom three non-orientable. Only the cylinder and mobius strip are not closed. 0 plane a 13- -----.. ---_-- b strip Figure 5-3 b An Introduction to Surface Topology 42 Chapt. 5 It turns out that every closed 2-manifold (whether orientable or not) can be represented in this manner. In fact (see Frgchet and Fan, [FFL] p . 63) we have the following theorem: Thm.

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