#### For citation:

Abrosimov M. B., Los I. V., Kostin S. V. The construction of all nonisomorphic minimum vertex extensions of the graph by the method of canonical representatives. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2021, vol. 21, iss. 2, pp. 238-245. DOI: 10.18500/1816-9791-2021-21-2-238-245

# The construction of all nonisomorphic minimum vertex extensions of the graph by the method of canonical representatives

A graph $G = (V, \alpha)$ is called \textit{primitive} if there exists a natural $k$ such that between any pair of vertices of the graph $G$ there is a route of length $k$. This paper considers undirected graphs with exponent 2. A criterion for the primitivity of a graph with the exponent 2 and a necessary condition are proved. A graph is primitive with the exponent 2 if and only if its diameter is 1 or 2, and each of its edges is included in a triangle. A computational experiment on the construction of all primitive homogeneous graphs with the number of vertices up to 16 and the exponent 2 is described, its results are analyzed. All homogeneous graphs of orders 2, 3, and 4, which are primitive with the exponent 2, are given, and for homogeneous graphs of order 5, the number of primitive graphs with the exponent 2 is determined.

- Frobenius F. G.
*Uber Matrizen aus nicht negativen Elementen.*Berlin, Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften, 1912. 22 p. - Wiеlandt H. Unzerlegbare nicht negative Matrizen.
*Mathematische Zeitschrift,*1950, vol. 52, pp. 642–648. https://doi.org/10.1007/BF02230720 - Fomichev V. M., Avezova Y. E., Koreneva A. M., Kyazhin S. N. Primitivity and local primitivity of digraphs and nonnegative matrices.
*Journal of Applied and Industrial Mathematics,*2018, vol. 12, iss. 3, pp. 453–469. https://doi.org/10.1134/S1990478918030067 - Jin M., Lee S. G., Seol H. G. Exponents of r-regular primitive matrices.
*Information Center for Mathematical Sciences,*2003, vol. 6, no. 2, pp. 51–57. - Bueno M. I., Furtado S. On the exponent of R-regular primitive matrices.
*ELA.**The Electronic Journal of Linear Algebra,*2008, vol. 17, pp. 28–47. https://doi.org/10.13001/1081- 3810.1247 - Kim B., Song B., Hwang W. Nonnegative primitive matrices with exponent 2.
*Linear Algebra and its Applications,*2005, no. 407, pp. 162–168. https://doi.org/10.1016/j.laa.2005.05.018 - Sachkov V. N., Oshkin I. B. Exponents of classes of non-negative matrices.
*Discrete Mathematics and Applications,*1993, vol. 3, iss. 4, pp. 365–375. - Salii V. N. Minimal primitive extensions of oriented graphs.
*Prikladnaya Diskretnaya Matematika,*2008. no. 1 (1), pp. 116–119 (in Russian). - Fomichev V. M. The estimates of exponents for primitive graphs.
*Prikladnaya Diskretnaya Matematika*, 2011, no. 2 (12), pp. 101–112 (in Russian). - Fomichev V. M., Avezova Y. E. The exact formula for the exponents of the mixing digraphs of register transformations.
*Journal of Applied and Industrial Mathematics,*2020, vol. 14, iss. 2, pp. 308–319. https://doi.org/10.1134/S199047892002009X - Meringer M. Fast generation of regular graphs and construction of cages.
*Journal of Graph Theory,*1999, vol. 30, iss. 2, pp. 137–146. https://doi.org/10.1002/(SICI)1097- 0118(199902)30:2<_x0031_37:_x003a_AID-JGT7>3.0.CO;2-G - Abrosimov M. B., Kostin S. V. About primitive regular graphs with exponent 2.
*Applied Discrete Mathematics.*Supplement, 2017, no. 10, pp. 131–134 (in Russian). https://doi.org/10.17223/2226308X/10/51 - Kostin S. V. On the use of graph theory problems for the intellectual development of students.
*Matematika v obrazovanii*[Mathematics in Education], 2014, iss. 10, pp. 68–74 (in Russian).

- 1183 reads