TY - JOUR

T1 - On small distance-regular graphs with the intersection arrays {mn - 1, (m - 1)(n + 1), n - M + 1; 1, 1, (m - 1)(n + 1)}

AU - Makhnev, A. A.

AU - Golubyatnikov, M. P.

N1 - The research was supported by the Russian Science Foundation (project № 19-71-10067).

PY - 2023

Y1 - 2023

N2 - 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. © 2023 Walter de Gruyter GmbH, Berlin/Boston.

AB - 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. © 2023 Walter de Gruyter GmbH, Berlin/Boston.

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U2 - 10.1515/dma-2023-0025

DO - 10.1515/dma-2023-0025

M3 - Article

VL - 33

SP - 273

EP - 281

JO - Discrete Mathematics and Applications

JF - Discrete Mathematics and Applications

SN - 0924-9265

IS - 5.1

ER -