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The generalized Petersen graphs also include the n-prism (,) the Dürer graph (,), the Möbius-Kantor graph (,), the dodecahedron (,), the Desargues graph (,) and the Nauru graph (,). The Petersen family consists of the seven graphs that can be formed from the Petersen graph by zero or more applications of Δ-Y or Y-Δ transforms .
The Petersen family. K 6 is at the top of the illustration, K 3,3,1 is in the upper right, and the Petersen graph is at the bottom. The blue links indicate ΔY- or YΔ-transforms between graphs in the family. In graph theory, the Petersen family is a set of seven undirected graphs that includes the Petersen graph and the complete graph K 6.
Coxeter's notation for the same graph would be {n} + {n/k}, a combination of the Schläfli symbols for the regular n-gon and star polygon from which the graph is formed. The Petersen graph itself is G(5, 2) or {5} + {5/2}. Any generalized Petersen graph can also be constructed from a voltage graph with two vertices, two self-loops, and one ...
A graph H is called a topological minor of a graph G if a subdivision of H is isomorphic to a subgraph of G. [21] Every topological minor is also a minor. The converse however is not true in general (for instance the complete graph K 5 in the Petersen graph is a minor but not a topological one), but holds for graph with maximum degree not ...
Complementarily, an intrinsically linked graph is a graph that does not have a linkless embedding. Flat embeddings are automatically linkless, but not vice versa. [2] The complete graph K 6, the Petersen graph, and the other five graphs in the Petersen family do not have linkless embeddings. [1]
The Petersen graph. The Petersen graph is an undirected graph with ten vertices and fifteen edges, commonly drawn as a pentagram within a pentagon, with corresponding vertices attached to each other. It has many unusual mathematical properties, and has frequently been used as a counterexample to conjectures in graph theory.
In particular, the spectrum of a highly symmetrical graph, such as the Petersen graph, has few distinct values [1] (the Petersen graph has 3, which is the minimum possible, given its diameter). For Cayley graphs, the spectrum can be related directly to the structure of the group, in particular to its irreducible characters. [1] [3]
See Families of sets for related families of non-graph combinatorial objects, graphs for individual graphs and graph families parametrized by a small number of numeric parameters, and graph theory for more general information about graph theory. See also Category:Graph operations for graphs distinguished for the specific way of their construction