More results on degree deviation and degree variance

Document Type : Research Paper

Authors

1 Department of Mathematics, Tafresh University, 39518-79611, Tafresh, Iran

2 Department of Mathematical Sciences, Yazd University, 89195-741, Yazd, Iran

Abstract

‎This paper investigates degree deviation and variance in graph theory‎, ‎with a specific focus on $k$-regular graphs and subdivision of graphs‎. ‎These metrics are fundamental for characterizing graph irregularity and have significant applications in network analysis and the social sciences‎.
‎Furthermore‎, ‎we introduce novel geometric measures of irregularity‎, ‎geometric degree deviation and geometric degree variance derived from the geometric mean of vertex degrees‎.
‎By means of rigorous theorems and illustrative examples‎, ‎we explore the relationships between graph structures and their degree properties‎. ‎Our findings seek to advance the current understanding of graph irregularity and provide a solid foundation for future research in this area‎.

Keywords

Main Subjects

References

[1] Abdo, H., Brandt, S., & Dimitrov, D. (2014). The total irregularity of graph. Discrete Math. Theor. Comput. Sci., 16, 201-206. https://doi.org/10.46298/dmtcs.1263.
[2] Albertson, M. O. (1997). The irregularity of a graph. Ars Combin, 46, 219-225.
[3] Alikhani, S., Ghanbari, N., & Soltani, S. (2023). Total dominator chromatic numbre of k-subdivision of graphs. Art Discrete Appl. Math, 6 #P1.10.
[4] Bell, F. K. (1992). A note on the irregularity of a graph. Linear Algebra and its Applications, 161, 45-54. https://doi.org/10.1016/0024-3795(92)90004-t.
[5] Collatz, L., & Sinogowitz, U. (1957). Spektren endlicher grafen. Abh. Math. sem. Univ. Hamburg, 21, 63-77. https://doi.org/10.1007/bf02941924.
[6] Furtula, B., Gutman, I., Vukicevic, Z. K., Lekishvili, G., & Popivoda, G. (2015). On an old/new degreebased topological index. Bull. Acad. Serbe Sci. Arts (Cl. Sci. Math. Natur.), 40, 19-31.
[7] Ghalavand, A., Reti, T., Milovanovic, I. Z., & Ashra , A. R. (2022) Graph irregularity characterization with particular regard to bidegreed graphs. arXiv, DOI: arXiv:2211.06744. https://doi.org/10.48550/arXiv.2211.06744.
[8] Gutman, I. (2016). Irregularity of molecular graph. Kragujevac J. Sci., 38, 99-109.
https://doi.org/10.5937/kgjsci1638071g.
[9] Nikiforov, V. (2006). Eigenvalues and degree deviation in graph. Linear Alg. Appl., 414, 347-360. https://doi.org/10.1016/j.laa.2005.10.011.
[10] West, D. B. (2001). Introduction to graph theory. 2nd ed. , Prentice Hall.
[11] Sayadi, M., Barzegar, H., Alikhani, S., & Ghanbari, N. (2025, April). Bell's Degree Variance and Degree Deviation in Graphs: Analyzing Optimal Graph Based on These Irregularity Measures, Control and Optimization in Applied Mathematics, Available Online. https://doi.org/10.30473/coam.2025.73746.1288. 
Journal of Mahani Mathematical Research
Volume 15, Issue 1 - Serial Number 33
January 2026
Pages 111-123

Files

History

  • Receive Date: 28 January 2025
  • Revise Date: 02 June 2025
  • Accept Date: 19 July 2025

Share

How to cite

Statistics