On Exponents Of Primitive Graphs

A connected gaph G is primitive provided there exists a positive integerk such that for each pair of vertices u and v in G there is a walk of lengtht that connects u and v. The smallest of such positive integers k is calledthe exponent of G and is denoted by exp(G). In this paper, we give a newbound on exponent of primitive graphs G in terms of the length of thesmallest cycle of G. We show that the new bound is sharp andgeneralizes the bounds given by Shao and Liu et. al

Synchronizing Automata on Quasi Eulerian Digraph

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