Geodesic
About the maximum number of vertices in primitive regular graphs with exponent equals $3$
Prikladnaâ diskretnaâ matematika, no. 1 (2025), pp. 98-109

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Some results on the maximum number of vertices in primitive regular graphs with exponent $3$ are presented. We have found upper bound of this number depending on the degree $p: n_p \le p^3-p^2-3p+5$. Also, the exact value of the maximum number of vertices in primitive cubic graphs with exponent $3$ is given: $n_3 = 12$. A computation experiment has been conducted, and we have found the number of primitive regular graphs with degree $p \le 9$, number of vertices $n \le 16$ and exponent $3$ for each $(n,p)$ pair.
Keywords: primitive graph, regular graph, the maximum number of vertices. 
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     author = {I. V. Los and M. B. Abrosimov},
     title = {About the maximum number of vertices in primitive regular graphs with exponent equals $3$},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {98--109},
     publisher = {mathdoc},
     number = {1},
     year = {2025},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2025_1_a5/}
}
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I. V. Los; M. B. Abrosimov. About the maximum number of vertices in primitive regular graphs with exponent equals $3$. Prikladnaâ diskretnaâ matematika, no. 1 (2025), pp. 98-109. http://geodesic.mathdoc.fr/item/PDM_2025_1_a5/