I'm trying to find the cycle index of the Petersen graph $G$. Therefore I need all the symmetries of it. Let's say that we call our vertices $1$ to $10$, where $1$ is the top vertex, then $2$ to $5$ the ones on the outer pentagon (counter-clockwise), then $6$ is the top vertex of the star in the middle, and then $7$ to $10$ the rest (counter-clockwise as well).

First I searched all symmetries that fix the vertex $1$. I found $12$ of them, and summing up their respective monomials for the cycle index, I get:

$$x_1^{10} + 4x_1^{4}x_2^{3} + 3x_1^{2}x_2^{4} + 2x_1x_3^{3} + 2x_1x_3x_6$$

By the orbit-stabiliser theorem, if I call $G_1$ the group of symmetries that fix $1$, I get:

$$|G| = |Orb(1)| \cdot |G_1| = 10 \cdot 12 = 120$$

So I've got $120$ symmetries to consider in total, which matches with what is written in the following link:

permutation group and cycle index question regarding peterson graph

Now I also get that therefore $G$ is bijective to $S_5$, but I don't know how that helps me finding the rest of the terms for the cyclic index and their coefficients.


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