Nonisotropic symplectic graphs over finite commutative rings

Songpon Sriwongsa; Siripong Sirisuk; Songpon Sriwongsa; Siripong Sirisuk

doi:10.3934/math.2022049

AIMS Mathematics

2022, Volume 7, Issue 1: 821-839. doi: 10.3934/math.2022049

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Research article

Nonisotropic symplectic graphs over finite commutative rings

Songpon Sriwongsa ¹,
Siripong Sirisuk ^{2
,
,}

1.
Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand
2.
Department of Mathematics and Statistics, Faculty of Science and Techonology, Thammasat University, Pathum Thani 12120, Thailand

Received: 04 June 2021 Accepted: 13 October 2021 Published: 18 October 2021
MSC : 05C25, 13H05

In this paper, we study two types of nonisotropic symplectic graphs over finite commutative rings defined by nonisotropic free submodules of rank $ 2 $ and McCoy rank of matrices. We prove that the graphs are quasi-strongly regular or Deza graphs and we find their parameters. The diameter and vertex transitivity are also analyzed. Moreover, we study subconstituents of these nonisotropic symplectic graphs.
- local ring,
- McCoy rank, nonisotropic subspace,
- symplectic space
Citation: Songpon Sriwongsa, Siripong Sirisuk. Nonisotropic symplectic graphs over finite commutative rings[J]. AIMS Mathematics, 2022, 7(1): 821-839. doi: 10.3934/math.2022049

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Abstract

In this paper, we study two types of nonisotropic symplectic graphs over finite commutative rings defined by nonisotropic free submodules of rank $ 2 $ and McCoy rank of matrices. We prove that the graphs are quasi-strongly regular or Deza graphs and we find their parameters. The diameter and vertex transitivity are also analyzed. Moreover, we study subconstituents of these nonisotropic symplectic graphs.

References

[1]	X. L. Hubaut, Strongly regular graphs, Discrete Math., 13 (1975), 357–381.
[2]	A. E. Brouwer, J. H. van Lint, Strongly regular graphs and partial geometries, in Enumeration and design (eds. D. H. Jackson and S. A. Vanstone), Academic Press, (1984), 85–122.
[3]	Z. Tang, Z. Wan, Symplectic graphs and their automorphisms, European J. Combin., 27 (2006), 38–50. doi: 10.1016/j.ejc.2004.08.002. doi: 10.1016/j.ejc.2004.08.002
[4]	C. Godsil, G. Royle, Chromatic number and the 2-rank of a graph, J. Combin. Theory Ser. B, 81 (2001), 142–149. doi: 10.1006/jctb.2000.2003. doi: 10.1006/jctb.2000.2003
[5]	C. Godsil, G. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics, vol. 207, Springer-Verlag, 2001.
[6]	J. J. Rotman, Projective planes, graphs, and simple algebras, J. Algebra, 155 (1993), 267–289. doi: 10.1006/jabr.1993.1044. doi: 10.1006/jabr.1993.1044
[7]	J. J. Rotman, P. M. Weichsel, Simple Lie algebras and graphs, J. Algebra, 169 (1994), 775–790. doi: 10.1006/jabr.1994.1307. doi: 10.1006/jabr.1994.1307
[8]	Y. Meemark, T. Prinyasart, On symplectic graphs modulo $p^n$, Discrete Math., 311 (2011), 1874–1878. doi: 10.1016/j.disc.2011.05.005. doi: 10.1016/j.disc.2011.05.005
[9]	Y. Meemark, T. Puirod, Symplectic graphs over finite local rings, European J. Combin., 34, (2013), 1114–1124. doi: 10.1016/j.ejc.2013.03.003. doi: 10.1016/j.ejc.2013.03.003
[10]	Y. Meemark, T. Puirod, Symplectic graphs over finite commutative rings, European J. Combin., 41 (2014), 298–307. doi: 10.1016/j.ejc.2014.05.004. doi: 10.1016/j.ejc.2014.05.004
[11]	S. Sirisuk, Y. Meemark, Generalized symplectic graphs and generalized orthogonal graphs over finite commutative rings, Linear Multilinear Algebra, 67 (2019), 2427–2450. doi: 10.1080/03081087.2018.1494124. doi: 10.1080/03081087.2018.1494124
[12]	L. Zeng, Z. Chai, R. Feng, C. Ma, Full automorphism group of the generalized symplectic graph, Sci. China Math., 56 (2013), 1509–1520. doi: 10.1007/s11425-013-4651-8. doi: 10.1007/s11425-013-4651-8
[13]	F. Li, K. Wang, J. Guo, Symplectic graphs modulo $pq$, Discrete Math., 313 (2013), 650–655. doi: 10.1016/j.disc.2012.12.002. doi: 10.1016/j.disc.2012.12.002
[14]	L. Su, J. Nan, Y. Wei, Construction of symplectic graphs by using $2$-dimensional symplectic nonisotropic subspaces over finite fields, Discrete Math., 343 (2020), 111689. doi: 10.1016/j.disc.2019.111689. doi: 10.1016/j.disc.2019.111689
[15]	N. H. McCoy, Rings and ideals, Carus Math. Monogr. No. 8, Math. Assoc. of Am., 1948.
[16]	J. V. Brawley, L. Carlitz, Enumeration of matrices with prescribed row and column sums, Linear Algebra Appl., 6 (1973), 165–174. doi: 10.1016/0024-3795(73)90016-5. doi: 10.1016/0024-3795(73)90016-5
[17]	D. Bollman, H. Ramirez, On the enumeration of matrices over finite commutative rings, Amer. Math. Monthly, 76 (1969), 1019–1023. doi: 10.2307/2317127. doi: 10.2307/2317127

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