Geodesic
The maximum number of vertices of primitive regular graphs of orders $2, 3, 4$ with exponent~$2$
Prikladnaâ diskretnaâ matematika, no. 2 (2021), pp. 97-104.

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In 2015, the results were obtained for the maximum number of vertices $ n_k $ in regular graphs of a given order $ k $ with a diameter $2$: $n_2 = 5$, $n_3 = 10$, $n_4 = 15$. In this paper, we investigate a similar question about the largest number of vertices $ np_k $ in a primitive regular graph of order $ k $ with exponent $2$. All primitive regular graphs with exponent $2$, except for the complete one, also have diameter $d =2 $. The following values were obtained for primitive regular graphs with exponent $2$: $np_2 = 3$, $np_3 = 4$, $np_4 = 11$.
Keywords: primitive graph, exponent, regular graph.
Mots-clés : primitive matrix 
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M. B. Abrosimov; S. V. Kostin; I. V. Los. The maximum number of vertices of primitive regular graphs of orders $2, 3, 4$ with exponent~$2$. Prikladnaâ diskretnaâ matematika, no. 2 (2021), pp. 97-104. http://geodesic.mathdoc.fr/item/PDM_2021_2_a5/

[1] Wielandt H., “Unzerlegbare nicht negative Matrizen”, Math. Zeitschr., 52 (1950), 642–648

[2] Sachkov V. N., Oshkin I. B., “Exponents of classes of non-negative matrices”, Diskretnaya Matematika, 1993, no. 2, 150–159 (in Russian)

[3] Salii V. N., “Minimal primitive extensions of oriented graphs”, Prikladnaya Diskretnaya Matematika, 2008, no. 1(1), 116–119 (in Russian)

[4] Fomichev V. M., “The estimates of exponents for primitive graphs”, Prikladnaya Diskretnaya Matematika, 2011, no. 2(11), 101–112 (in Russian)

[5] Fomichev V. M., Avezova Ya. E., “The exact formula for the exponents of the mixing digraphs of register transformations”, J. Appl. Ind. Math., 2020, no. 14, 308–320

[6] Jin M., Lee S. G., Seol H. G., “Exponents of $r$-regular primitive matrices”, Inform. Center Math. Sci., 6:2 (2003), 51–57

[7] Bueno M. I., Furtado S., “On the exponent of $r$-regular primitive matrices”, ELA. Electronic J. Linear Algebra, 17 (2008), 28–47

[8] Kim B., Song B., Hwang W., “Nonnegative primitive matrices with exponent 2”, Linear Algebra Appl., 2005, no. 407, 162–168

[9] Hoa V. D., Do M. T., “$k$-Regular graph with diameter 2”, Int. J. Adv. Comput. Technol., 4:5 (2015), 14–19

[10] Los' I. V., Abrosimov M. B., Kostin S. V., “On the question of primitive regular graphs with exponent equals to 2 and 3”, Komp'yuternye Nauki i Informatsionnye Tekhnologii, Nauka Publ., Saratov, 2018, 251–253 (in Russian)

[11] Kostin S. V., “On the use of graph theory problems for the intellectual development of students”, Matematika v Obrazovanii, ed. I. S. Emel'yanova, Chuvash University Publ., Cheboksary, 2014, 68–74 (in Russian)