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.Comment: 8 pages, 1 figur