Die Gruppe des Petersenschen Graphen und der Kantensysteme der regul�ren Polyeder

Vertex-transitive graphs

“…By [5] p. 194, --~ C5 is the well-known Petersen graph P (cf. also [3]). According to [4], p. 11, R. M. Foster proved that P is not a Cayley graph, i.e., dev P ~ 2.…”

The groups of the generalized Petersen graphs

“…An alternative presentation of .4(10,2) (see (1), p. 167) requires defining A on F(#(10,2)) to have cycle structure A = («", V 2 , V B ) (U v V t , Mg) (U 2 , V 6 , U 9 ) (U 3 , U 6 , V 9 ) (U A , U 7 , Vj) (U 5 , V 7 , V 3 ) whence A(10, 2) = </o,A>with p 10 = A 3 = ( A/9 2)2 = p 5 A p -5 A -l = I One sees that S = (pA) 2 pA-x p" 2 -When k 2 = 1 (modn), we have the graphs G(4,1), G (8, 3), G(12, 5), and #(24, 5). One verifies that <x defined on V(G(n, k)) in these four cases by = v iU is an element of A(n, k) clearly not in B(n, k).…”

“…There are just two types of 7-circuits in (? (13,5). They are represented byZ x = [«", u lt u 2 , u 3 ,v 3 , v a , v 0 ] and Z 2 = [u 0 , u x , u 2 , v 2 , v 10 , v 6 , v 0 ].…”

A Note on the Generalized Petersen Graphs That Are Also Cayley Graphs

“…It is known [4,5] that A(10, 2)$Alt 5 _Z 2 , where Alt 5 is the alternating group of degree 5. There are 24 rotations around six 5-axes (through the central points of antipodal pentagons), 20 around ten 3-axes (through antipodal vertices) and 15 around 15 2-axes (through the central points of antipodal edges) in A(10, 2).…”