Geodesic
The estimates of exponents for primitive graphs
Prikladnaâ diskretnaâ matematika, no. 2 (2011), pp. 101-112

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The estimates of exponents of $n$-vertex primitive digraphs and undirected graphs are improved. The digraphs considered contain two prime contours with coprime lengths $l$ and $\lambda$. For them, accessible estimates of the order $\mathrm O(\max\{l\lambda,f(l,\lambda,n)\}$ are obtained, where $f(l,\lambda,n)$ is a linear polynomial. The exponent of undirected graph is no more $2n-l-1$, where $l$ is the length of the longest cycle with odd length in graph. Primitive digraphs with maximal exponent ($n^2-2n+2$, H. Wielandt, 1950) and undirected graphs with maximal exponent ($2n-2$) are completely described.
Keywords: graph exponent, primitive graphs. 
@article{PDM_2011_2_a8,
     author = {V. M. Fomichev},
     title = {The estimates of exponents for primitive graphs},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {101--112},
     publisher = {mathdoc},
     number = {2},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2011_2_a8/}
}
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V. M. Fomichev. The estimates of exponents for primitive graphs. Prikladnaâ diskretnaâ matematika, no. 2 (2011), pp. 101-112. http://geodesic.mathdoc.fr/item/PDM_2011_2_a8/