Petersen graph

Figure 3.1 Smallest spherical, toroidal, Klein bottle and projective fullerenes. The first column lists the graphs drawn m the plane, the second the map on the appropriate surface and the third the dual on the same surface. The examples are (a) Dodecahedron (dual Icosahedron), (b) the Heawood graph (dual Ky), (c) a smallest Klein bottle polyhex (dual 3,3,3), and (d) the Petersen graph (dual Ke).

In this terminology, our definition of projective fullerenes amounts to selection of cell-complex projective-planar 3-valent maps with only 5- and 6-gonal feces. As noted above, P5 — 6 for these maps. Thus, the Petersen graph is die smallest projective fullerene. In general, the projective fullerenes are exactly the antipodal quotients of the centrally symmetric spherical fullerenes. 

HoSh93] D. A. Holton and J. Sheehan, The Petersen Graph, Cambridge University Press, 1993. 

Fig. 3. The four-vertex Petersen graph representing the tunneling paths between the trigonal wells in the T t2 case. The arrows show the direction of probability flux from the corresponding wells.

Fig. 4. The Petersen graph for the topology of the G 0 (g 0 h) JT surfaces for preferential H coupling by Ceulemans and Fowler [29]. The numbered 10 vertices of the graph represent the 10 trigonal minima (ft orbits). The centers of the edges represent the D2 saddles (8 orbits). There are two types of tunneling between wells one between adjacent wells (minima connected with edges in the graph) the other between non-adjacent wells.

When we examine the Petersen-graph [23], we will discover 12 shortest cycles of length five (Table 4). Six of these cycles will be realized as faces in the embedding... 

Fig. 5. Induced embedding of the Petersen graph (left) and its complement (right).

Table 4. List of the twelve 5-cycles of the Petersen graph...

Table 5. List of the fifteen faces of the complement of the Petersen graph given in Fig. 5b...

The Euler genus of the complement of the Petersen-graph therefore has to be greater than or equal to two ... 

M. Randic, A systematic study of symmetry properties of graphs. I. Petersen graph, Croat. Chem. Acta 49 (1977) 643-655. 

Fig. 1 Three graphs with automorphism group S5 a the complete 5-graph, K5 b the (10-vertex) Petersen graph c an extended K5 graph with 20 vertices...

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