A family of generalized strongly regular graphs of grade 2

doi:10.3868/S140-DDD-023-001-X

Front. Math. China

2023, Vol. 18

Issue (1) : 33-42 https://doi.org/10.3868/S140-DDD-023-001-X

RESEARCH　ARTICLE

A family of generalized strongly regular graphs of grade 2

Simin SONG¹, Lifang YANG², Gengsheng ZHANG^1,³(

)

¹. College of Mathematics Science, Hebei Normal University, Shijiazhuang 050024, China
². Department of Basic Education, Shijiazhuang Engineering Vocational College, Shijiazhuang 050061, China
³. Hebei Key Laboratory of Computational Mathematics and Applications, Shijiazhuang 050024, China

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Abstract

A generalized strongly regular graph of grade $p$ , as a new generalization of strongly regular graphs, is a regular graph such that the number of common neighbours of both any two adjacent vertices and any two non-adjacent vertices takes on $p$ distinct values. For any vertex $v$ of a generalized strongly regular graph of grade 2 with parameters $(n, k; a 1, a 2; c 1, c 2)$ , if the number of the vertices that are adjacent to $v$ and share $a i (i = 1, 2)$ common neighbours with $v$ , or are non-adjacent to $v$ and share $c i (i = 1, 2)$ common neighbours with $v$ is independent of the choice of the vertex $v$ , then the generalized strongly regular graph of grade 2 is free. In this paper, we investigate the generalized strongly regular graph of grade 2 with parameters $(n, k; k − 1,$ $a 2; k − 1, c 2)$ and provide the sufficient and necessary conditions for the existence of a family of free generalized strongly regular graphs of grade 2.

Keywords Strongly regular graph generalized strongly regular graph graph composition, isomorphism

Corresponding Author(s): Gengsheng ZHANG

Online First Date: 22 May 2023 Issue Date: 31 May 2023

Cite this article:

Simin SONG,Lifang YANG,Gengsheng ZHANG. A family of generalized strongly regular graphs of grade 2[J]. Front. Math. China, 2023, 18(1): 33-42.

URL:

https://academic.hep.com.cn/fmc/EN/10.3868/S140-DDD-023-001-X
https://academic.hep.com.cn/fmc/EN/Y2023/V18/I1/33

Fig.1 GSRG(10,5;4,2;4,2)

Fig.2 GSRG(8,5;3,2;5,4)

Fig.3 GSRG(20,9;8,2;8,4)

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