I have a problem where I have a graph $\Gamma$ and it's automorphism group $G$.
I look at the cycle index of $Z(G,V(\Gamma))$
And then I substitute $x_i$ with $(1+x^i).$
I get a polynomial in the variable $x$
What is the meaning of the coefficients of $x^i$ in said polynomial?
If it helps, the particulars are:
$\Gamma = Cay(Z_{13},${$1,3,4,9,10,12$}$)$
$Z(G,Z_{13}) = 1/78(x_1^{13}+13x_1x_2^6+26x_1x_3^4+26x_1x_6^2+12x_{13})$
$Z(G,1+x)= 1+x+2x^2+6x^3+13x^4+19x^5+28x^6+28x^7+19x^8+13x^9+6x^{10}+2x^{11}+x^{12}+x^{13}$
I suspect it has something to do with orbits of induced subgraphs but still needs clarifications.
Thanks in advnace,
Shay