"The" graph is the path graph on two vertices: .An -graph for and is a generalization of a generalized Petersen graph and has vertex setand edge setwhere the subscripts are read modulo (Bouwer et al. 1988, itnik et al. ). Such graphs can be constructed by graph expansion on .If the restriction is relaxed to allow and to equal , gives the ladder rung graph and gives the sunlet graph .Two -graphs and are isomorphic iff there exists an integer relatively prime to such that either or (Boben et al. 2005, Horvat et al. 2012, itnik 2012).The graph is connected iff . If , then the graph consists of copies of (itnik et al. 2012).The -graph corresponds to copies of the graph The following table summarizes special named -graphs and classes of named -graphs.graphcubical graph Petersen graph Dürer graphMöbius-Kantor graphdodecahedral graphDesargues graphNauru graphprism graph generalized Petersen graph All -graphs..
The dodecahedral graph is the Platonic graph corresponding to the connectivity of the vertices of a dodecahedron, illustrated above in four embeddings. The left embedding shows a stereographic projection of the dodecahedron, the second an orthographic projection, the third is from Read and Wilson (1998, p. 162), and the fourth is derived from LCF notation.It is the cubic symmetric denoted and is isomorphic to the generalized Petersen graph . It can be described in LCF notation as [10, 7, 4, , , 10, , 7, , .It is distance-regular with intersection array and is also distance-transitive.It is also a unit-distance graph (Gerbracht2008), as shown above in a unit-distance drawing.Finding a Hamiltonian cycle on this graph is known as the icosian game. The dodecahedral graph is not Hamilton-connected and is the only known example of a vertex-transitive Hamiltonian graph (other than cycle graphs ) that is not H-*-connected (Stan Wagon, pers...
The Petersen graph is the cubic graph on 10 vertices and 15 edges which is the unique -cage graph (Harary 1994, p. 175), as well as the unique -Moore graph. It can be constructed as the graph expansion of with steps 1 and 2, where is a path graph (Biggs 1993, p. 119). Excising an edge of the Petersen graph gives the 4-Möbius ladder . It is illustrated above in several embeddings (D'Angelo and Saaty and Kainen 1986; Harary 1994, p. 89; West 2000, p. 229; Knuth 2008, p. 39).The Petersen graph can be generalized, with the resulting graphs being known as generalized Petersen graphs for and . The Petersen graph corresponds to .The Petersen graph has girth 5, diameter 2, edge chromatic number 4, chromatic number 3, and chromatic polynomialThe Petersen graph is a cubic symmetric graph and is nonplanar. The following elegant proof due to D. West demonstrates that the Petersen graph is nonhamiltonian. If there is a..
The generalized Petersen graph , also denoted (Biggs 1993, p. 119; Pemmaraju and Skiena 2003, p. 215), for and is a connected cubic graph consisting of an inner star polygon (circulant graph ) and an outer regular polygon (cycle graph ) with corresponding vertices in the inner and outer polygons connected with edges. These graphs were introduced by Coxeter (1950) and named by Watkins (1969).Since the generalized Petersen graph is cubic, , where is the edge count and is the vertex count. More specifically, has nodes and edges. Generalized Petersen graphs are implemented in the Wolfram Language as PetersenGraph[k, n] and their properties are available using GraphData["GeneralizedPetersen", k, n].Generalized Petersen graphs may be further generalized to Igraphs.For odd, is isomorphic to . So, for example, , , , , and so on. The numbers of nonisomorphic generalized Petersen graphs on , 8, ... nodes are 1, 1, 2, 2, 2, 3,..