Pfaffian and Computational Analysis of Determinants ofSkew-Symmetric Matrices for HomogeneousTournaments: Pfaffian and Computational Analysis of Determinants of Skew-Symmetric Matrices for Homogeneous Tournaments

R.N. Ganikhodzhaev; M.A. Tadzhieva; S.V. Maqsimova

doi:10.56143/ujmcs.v1i2.19

Authors

R.N. Ganikhodzhaev National University of Uzbekistan Author https://orcid.org/0000-0001-6551-5257
M.A. Tadzhieva Tashkent state transport university Author https://orcid.org/0000-0001-9232-3365
S.V. Maqsimova Andijan State University Author https://orcid.org/0009-0007-0083-0978

DOI:

https://doi.org/10.56143/ujmcs.v1i2.19

Abstract

This paper investigates the properties of skew-symmetric matrices with a focus on the case when
the order m=6. After discussing the fundamental characteristics of skew-symmetric matrices,
we derive the structure of their determinants for even values of m, particularly when m=2 and
4, as preliminary cases to support our main study of the case m= 6. Furthermore, we introduce
a tournament representation of such matrices, linking matrix entries to directed graphs based
on their signs. We studied all non-isomorphic tournaments of order 6 and identified six of them
as homogeneous, computed their corresponding matrices and determinants.

Author Biographies

R.N. Ganikhodzhaev , National University of Uzbekistan

Address: Doctor of physical and mathematical sciences, professor of the department of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan
M.A. Tadzhieva, Tashkent state transport university

Address: PhD, head of the department of higher mathematics, Tashkent State Transport University, Institute of Mathematics named after V.I. Romanovsky Academy of Sciences of the Republic of Uzbekistan, Senior Researcher. Tashkent, Uzbekistan.
e-mail: mohbonut@mail.ru
ORCID ID: https://orcid.org/0000-0001-9232-3365
S.V. Maqsimova, Andijan State University

Address: Doctorate student of the department of Mathematics, Andijan State University, Andijan, Uzbekistan.
e-mail: hayitovas20@gmail.com
ORCID ID: https://orcid.org/0009-0007-0083-0978

References

[1] Kasteleyn, P. W. Graph Theory and Crystal Physics. In Graph Theory and Theoretical Physics. (1967).

[2] Kasteleyn, P. W. Dimer Statistics and Phase Transitions. Journal of Mathematical Physics. 4(2), p.287–293,(1963).

[3] Moon,J. W. Topics on Tournaments, New York: Holt, Rinehart and Winston, p. 112. (1968).

[4] Horn, R. A., & Johnson, C. R. Matrix Analysis. Cambridge University. (2012).

[5] Harary F., Palmer E.M. Graphical enumeration. Academic Press New York and London. (1973).

[6] G. Chartrand and H. Jordon and V. Vatter and P. Zhang. Graphs and Digraphs. CRC Press. p. 364. (2024).

[7] Koh Kh., Dong F., and Tay E.G. Introduction to graph theory. World Scientific. p. 308. (2024).

[8] Ganikhodzhaev R.N. Quadratic stochastic operators, Lyapunov function and tournaments, Acad. Sci. Sb. Math., 76(2), p. 489-506. (1993)

[9] Ganikhodzhaev R.N., Tadzhieva M.A. Stability of fixed points of discrete dynamic systems of Volterra type. AIP Conference Proceedings, 2021. V. 2365. P. 060005-1, 060005-7. https://doi.org/10.1063/5.0057979. .

[10] Ganikhodzhaev R.N., Tadzieva M.A., Eshmamatova D.B. Dynamical Proporties of Quadratic Homeomorphisms of a Finite-Dimensional Simplex. Journal of Mathematical Sciences, 245(3). P. 398-402.

Pfaffian and Computational Analysis of Determinants ofSkew-Symmetric Matrices for HomogeneousTournaments