ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

#### 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

Published online:
31.05.2021
Full text:
Language:
Russian
Article type:
Article
UDC:
519.17

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

Autors:
Abrosimov Mikhail Borisovich, Saratov State University
Los Ilya V., Saratov State University
Abstract:

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.

Key words:
Acknowledgments:
This work was supported by the Ministry of Science and Higher Education of the Russian Federation in the framework of the state task (project No. FSRR-2020-0006).
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