The improvement of exponent's estimates for primitive graphs

The estimates of exponents of $n$-vertex primitive digraphs are improved. The digraphs considered contain two prime contours whose lengths $l$ and $\lambda$ are coprime numbers. Accessible estimates of the order $O(\max\{l\lambda,f(l,\lambda,n)\})$ are obtained, where $f(l,\lambda,n)$ is a linear polynomial. Primitive digraphs whose exponents are maximal ($n^2-2n+2$, H. Wielandt, 1950), are described completely. The estimates of exponents of $n$-vertex primitive undirected graphs are improved too. In particular, the exponent of an undirected graph is no more $2n-l-1$ where $l$ is the length of the longest cycle with odd length in graph. Primitive undirected graphs whose exponents are maximal ($2n-2$) are described completely.