  Let $G$ be a finite group of order $n$, and
$Z_G=\mathbb{Z}\langle\zeta_{i,g}\mid g\in G,\ i=1,2,\dots,n\rangle$ be the
free generic algebra, with canonical action of $G$ according to
$(\zeta_{i,g})^x=\zeta_{i,x^{-1}g}$. It is proved that there exists a positive
integer $\upsilon(G)$ such that for any $g_1,g_2,\dots, g_{n}\in G$ $$
\upsilon(G)\cdot \zeta_{1,g_1}\zeta_{2,g_2}\dots\zeta_{n,g_{n}}=\sum_{i=1}^N
\gamma_i a_i\mathbf{tr}_G(b_i)c_i, $$ where $\gamma_1,\gamma_2,\dots,\gamma_N$
are integers, and $a_i, b_i, c_i$ are monomials in $\zeta_{i,g}$ such that
${\rm deg}(b_i)>0$ and ${\rm deg}(a_i)+{\rm deg}(b_i)+{\rm deg}(c_i)=n$. As a
consequence, if $R$ is a ring (not necessarily unital) acted on by $G$, then
the product $\upsilon(G)\cdot R^{n}$ is contained in the ideal
$\langle\mathbf{tr}_G(R)\rangle$ generated by all traces
$\mathbf{tr}_G(r)=\sum\limits_{g\in G}r^g$, $r\in R$. This gives the best
possible nilpotence bound in Bergman-Isaacs theorem for finite group actions on
non-commutative rings, which was a long standing problem. The main result was
obtained by transferring the problem to certain family of Cayley graphs, and
estimating their minimal eigenvalues by the clique numbers. It is proved that
the clique number $\omega(\Gamma)$ of any $k$-regular graph $\Gamma$ admits the
Delsarte upper bound $\omega(\Gamma)\leqslant\lfloor1-k/\lambda_{\rm
min}\rfloor$.
