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Symmetries

The Petersen graph is strongly regular (with signature srg(10,3,0,1)). It is also symmetric, meaning that it is edge transitive and vertex transitive. More strongly, it is 3-arc-transitive: every directed three-edge path in the Petersen graph can be transformed into every other such path by a symmetry of the graph. It is one of only 13 cubic distance-regular graphs.

The automorphism group of the Petersen graph is the symmetric group S 5 {\displaystyle S_{5}}; the action of S 5 {\displaystyle S_{5}} on the Petersen graph follows from its construction as a Kneser graph. The Petersen graph is a core: every homomorphism of the Petersen graph to itself is an automorphism. As shown in the figures, the drawings of the Petersen graph may exhibit five-way or three-way symmetry, but it is not possible to draw the Petersen graph in the plane in such a way that the drawing exhibits the full symmetry group of the graph.

Despite its high degree of symmetry, the Petersen graph is not a Cayley graph. It is the smallest vertex-transitive graph that is not a Cayley graph.