On small distance-regular graphs with the intersection arrays {mn − 1, (m − 1)(n + 1), n − m + 1; 1, 1, (m − 1)(n + 1)}
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Aleksandr A. Makhnev
and
Mikhail P. Golubyatnikov
Abstract
Let Γ be a diameter 3 distance-regular graph with a strongly regular graph Γ3, where Γ3 is the graph whose vertex set coincides with the vertex set of the graph Γ and two vertices are adjacent whenever they are at distance 3 in the graph Γ. Computing the parameters of Γ3 by the intersection array of the graph Γ is considered as the direct problem. Recovering the intersection array of the graph Γ by the parameters of Γ3 is referred to as the inverse problem. The inverse problem for Γ3 has been solved earlier by A. A. Makhnev and M. S. Nirova. In the case where Γ3 is a pseudo-geometric graph of a net, a series of admissible intersection arrays has been obtained: {c2(u2 − m2) + 2c2m − c2 − 1, c2(u2 − m2), (c2 − 1)(u2 − m2) + 2c2m − c2; 1, c2, u2 − m2} (A. A. Makhnev, Wenbin Guo, M. P. Golubyatnikov). The cases c2 = 1 and c2 = 2 have been examined by A. A. Makhnev, M. P. Golubyatnikov and A. A. Makhnev, M. S. Nirova, respectively.
In this paper in the class of graphs with the intersection arrays {mn − 1, (m − 1)(n + 1)}, {n − m + 1}; 1, 1, (m − 1)(n + 1)} all admissible intersection arrays for {3 ≤ m ≤ 13} are found: {20,16,5; 1, 1,16}, {39,36,4; 1, 1,36}, {55,54,2; 1, 2,54}, {90,84,7; 1, 1,84}, {220,216,5; 1, 1,216}, {272,264,9; 1, 1,264} and {350,336,15; 1, 1,336}. It is demonstrated that graphs with the intersection arrays {20,16,5; 1, 1,16}, {39,36,4; 1, 1,36} and {90,84,7; 1, 1,84} do not exist.
Note: Originally published in Diskretnaya Matematika (2022) 34,№ 1, 76–87 (in Russian).
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Funding: The research was supported by the Russian Science Foundation (project № 19-71-10067).
References
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Articles in the same Issue
- Frontmatter
- On small distance-regular graphs with the intersection arrays {mn − 1, (m − 1)(n + 1), n − m + 1; 1, 1, (m − 1)(n + 1)}
- On algebraicity of lattices of ω-fibred formations of finite groups
- On polynomial-modular recursive sequences
- Classes of piecewise-quasiaffine transformations on the generalized 2-group of quaternions
- Limit theorem for a smoothed version of the spectral test for testing the equiprobability of a binary sequence
- Limit theorem for stationary distribution of a critical controlled branching process with immigration
Articles in the same Issue
- Frontmatter
- On small distance-regular graphs with the intersection arrays {mn − 1, (m − 1)(n + 1), n − m + 1; 1, 1, (m − 1)(n + 1)}
- On algebraicity of lattices of ω-fibred formations of finite groups
- On polynomial-modular recursive sequences
- Classes of piecewise-quasiaffine transformations on the generalized 2-group of quaternions
- Limit theorem for a smoothed version of the spectral test for testing the equiprobability of a binary sequence
- Limit theorem for stationary distribution of a critical controlled branching process with immigration
