  In 1964 \v{C}ern\'{y} conjectured that each $n$-state synchronizing automaton
posesses a reset word of length at most $(n-1)^2$. From the other side the best
known upper bound on the reset length (minimum length of reset words) is cubic
in $n$. Thus the main problem here is to prove quadratic (in $n$) upper bounds.
Since 1964, this problem has been solved for few special classes of \sa. One of
this result is due to Kari \cite{Ka03} for automata with Eulerian digraphs. In
this paper we introduce a new approach to prove quadratic upper bounds and
explain it in terms of Markov chains and Perron-Frobenius theories. Using this
approach we obtain a quadratic upper bound for a generalization of Eulerian
automata.
