Petersen Graph


The Petersen graph is the cubic graph on 10 vertices and 15 edges which is the unique (3,5)-cage graph (Harary 1994, p. 175), as well as the unique (3,5)-Moore graph. It can be constructed as the graph expansion of 5P_2 with steps 1 and 2, where P_2 is a path graph (Biggs 1993, p. 119). Excising an edge of the Petersen graph gives the 4-Möbius ladder Y_3. 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 GP(n,k) for n>=3 and k<n/2. The Petersen graph corresponds to GP(5,2).

The Petersen graph has girth 5, diameter 2, edge chromatic number 4, chromatic number 3, and chromatic polynomial


The 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 10-cycle C, then the graph consists of C plus five chords. If each chord joins vertices opposite on C, then there is a 4-cycle. Hence some chord e joins vertices at distance 4 along C. Now no chord incident to a vertex opposite an endpoint of e on C can be added without creating a cycle with at most four vertices. Therefore, the Petersen graph is nonhamiltonian. In fact, it is also the smallest hypohamiltonian graph.

The Petersen graph is one of two cubic graphs on 10 nodes with smallest possible graph crossing number of 2 (the other being an unnamed graph denoted CNG 2B by Pegg and Exoo 2009), making it a smallest cubic crossing number graph (Pegg and Exoo 2009, Clancy et al. 2019).

The Petersen graph is the unique almost Hamiltonian cubic graph on 10 vertices (Punnim et al. 2007). In fact, it is also maximally nonhamiltonian (Clark and Entringer 1983).

It is also a unit-distance graph (Gerbracht 2008).

The Petersen graph is the only smallest-girth graph which has no Tait coloring, It is the complement of the line graph of the complete graph K_5 (Skiena 1990, p. 139), and the odd graph O_3 (Skiena 1990, p. 162).

The Petersen graph is an integral graph with graph spectrum (-2)^41^53^1.

The bipartite double graph of the Petersen graph is the Desargues graph.


The line graph of the Petersen graph is the quartic vertex-transitive graph Qt39, illustrated above in several embeddings.

The Petersen graph is depicted on the covers of both the journals Journal of Graph Theory and Discrete Mathematics. It is also the lower right graph depicted on the cover of Harary (1994).


The Petersen graph provides a 6-coloring of the projective plane.

The Petersen graph is implemented in the Wolfram Language as PetersenGraph[] and a number of precomputed properties are available via GraphData["PetersenGraph"].

The following table summarizes some properties of the Petersen graph.

automorphism group order120
characteristic polynomial(x-3)(x-1)^5(x+2)^4
chromatic number3
claw-free graphno
clique number2
graph complement name5-triangular graph
distance-regular graphyes
edge chromatic number4
edge connectivity3
edge count15
edge transitiveyes
Eulerian graphno
Hamiltonian graphno
Hamiltonian cycle count0
Hamiltonian path count240
hypohamiltonian graphyes
hypotraceable graphno
integral graphyes
independence number4
line graphno
line graph namequartic vertex-transitive graph Qt39
perfect graphno
perfect matching graphyes
planar graphno
polyhedral graphno
regular graphyes
square-free graphyes
strongly regular graph(10,3,0,1)
symmetric graphyes
traceable graphyes
triangle-free graphyes
vertex connectivity3
vertex count10
vertex transitiveyes

See also

Cage Graph, Generalized Petersen Graph, Girth, Hoffman-Singleton Graph, Hypohamiltonian Graph, Integral Graph, Kneser Graph, Moore Graph, Odd Graph, Petersen Family Graphs, Smallest Cubic Crossing Number Graph

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Cite this as:

Weisstein, Eric W. "Petersen Graph." From MathWorld--A Wolfram Web Resource.

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