Meaning of coefficients in Z(G,1+x)

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.

Do you really mean $Z_13$ and $x_1^13$, or do we need some TeXnicalities? – Gerry Myerson Jul 18 '11 at 12:30
Could you explain your notation a bit? What is $Z_13$? Do you mean $Z_{13}$, if so write Z_{13}. I presume $Cay$ refers to the Cayley graph of a group with respect to some generators, but then I have trouble seeing why you have the element $34$ in $Z_{13}$ Also, it might help to define what $Z(G,Z_13)$ means. – t.b. Jul 18 '11 at 12:31
$Z_13$ means Z_{13} and $x_1^13$ means x_1^{13}, presumably. – Did Jul 18 '11 at 12:37