مجله مدل‌سازی پیشرفته ریاضی

مجله مدل‌سازی پیشرفته ریاضی

HOSOYA POLYNOMIAL OF CAYLEY GRAPHS OF D2n AND Q2n WITH A FOCUS ON SPECIFIC GENERATING SETS

نوع مقاله : مقاله پژوهشی

نویسندگان
1 Department of Mathematics, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran
2 Department of Mathematics, Mahshahr branch, Islamic Azad University, Mahshahr, Iran
چکیده
Cayley graphs are powerful tools for analyzing algebraic groups through their combinatorial and topological properties. In this paper, we investigate the Hosoya polynomial of Cayley graphs associated with the Dihedral group \( D_{2n} \) and the Generalized Quaternion group \( Q_{2^n} \), considering specific generating subsets. General formulas for computing these polynomials are presented, and their connection with the Wiener index is examined. Our results show that both the Hosoya polynomial coefficients and the Wiener index exhibit exponential growth with the group order, reflecting the increasing complexity of the graphs. This study contributes to the understanding of distance-based invariants in algebraic graph theory and provides a foundation for further exploration in network analysis.
کلیدواژه‌ها
موضوعات

عنوان مقاله English

HOSOYA POLYNOMIAL OF CAYLEY GRAPHS OF D2n AND Q2n WITH A FOCUS ON SPECIFIC GENERATING SETS

نویسندگان English

Sara Shokrolllahi Yancheshmeh 1
Maryam Jahandideh Khangheshlaghi 2
1 Department of Mathematics, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran
2 Department of Mathematics, Mahshahr branch, Islamic Azad University, Mahshahr, Iran
چکیده English

Cayley graphs are powerful tools for analyzing algebraic groups through their combinatorial and topological properties. In this paper, we investigate the Hosoya polynomial of Cayley graphs associated with the Dihedral group \( D_{2n} \) and the Generalized Quaternion group \( Q_{2^n} \), considering specific generating subsets. General formulas for computing these polynomials are presented, and their connection with the Wiener index is examined. Our results show that both the Hosoya polynomial coefficients and the Wiener index exhibit exponential growth with the group order, reflecting the increasing complexity of the graphs. This study contributes to the understanding of distance-based invariants in algebraic graph theory and provides a foundation for further exploration in network analysis.

کلیدواژه‌ها English

Hosoya Polynomial
Wiener Index
Cayley Graph
Dihedral Group
Generalized Quaternion Group
[1] Ashrafi, A.R. and Shabani, H. (2009). The Hosoya polynomial of TUC4C8(S) nanotubes. Digest
Journal of Nanomaterials and Biostructures, 4(3), pp. 453–457.
[2] Ashrafi, S., Imran, M., Bokhary, S. A. U. and Akhter, S. (2022). The Wiener index, degree distance index and Gutman index of composite hypergraphs. Heliyon, 8(12), pp. 1–10.
doi:10.1016/j.heliyon.2022.e12382
[3] Biggs, N. (1993). Algebraic Graph Theory. 2nd ed. Cambridge: Cambridge University Press.
doi:10.1017/CBO9780511608704.
[4] Darafsheh, M.R. (2010). Computation of topological indices of some graphs. Acta Applicandae Mathematicae, 110(3), pp. 1225–1235. doi: 10.1007/s10440-009-9503-8
[5] Dehmer, M. and Mowshowitz, A. (2011). A history of graph entropy measures. Information Sciences,
181(1), pp. 57–78. doi:10.1016/j.ins.2010.08.041.
[6] Dobrynin, A. A., Entringer, R. and Gutman, I. (2001). Wiener index of trees: Theory and applications. Acta Applicandae Mathematicae, 66(3), pp. 211–249. doi: 10.1023/A:1010767517079.
[7] Dobrynin, A. A., Gutman, I., Klavˇzar, S. and Zigert, P. (2002). Wiener index of hexagonal systems. ˇ
Acta Applicandae Mathematicae, 72, pp. 247–294. doi: 10.1023/A:1016290123303.
[8] Godsil, C. and Royle, G. (2001). Algebraic Graph Theory. New York: Springer-Verlag. doi:
10.1007/978-1-4613-0163-9.
[9] Gutman, I. and Polansky, O. E. (1986). Mathematical Concepts in Organic Chemistry. Berlin:
Springer-Verlag. doi:10.1007/978-3-642-70982-1.
[10] Gutman, I. and Dobrynin, A. A. (1998). The Szeged index a success story. Graph Theory Notes of
New York, 34, pp. 37–44.
[11] Hosoya, H. (1971). Topological index. A newly proposed quantity characterizing the topological
nature of structural isomers of saturated hydrocarbons. Bulletin of the Chemical Society of Japan,
44(9), pp. 2332–2339. doi: 10.1246/bcsj.44.2332.
[12] Hosoya, H. (1988). On some counting polynomials in chemistry. Discrete Applied Mathematics, 19(1-
3), pp. 239–257. doi: 10.1016/0166-218X(88)90017-0.
[13] Moghani, G. A., Ashrafi, A. R., Naghdi, S. and Ahmadi, M. R. (2007). Automorphism groups of
some chemical graphs. In: Campbell, C. M., Quick, M. R., Robertson, E. F. and Smith, G. C. (eds.)
Groups St Andrews 2005. London Mathematical Society Lecture Note Series. Cambridge University
Press, pp. 630–639. doi:10.1017/CBO9780511721205.026.
[14] Shokrollahi Yancheshmeh, S., Modabbernia, R. and Jahandideh, M. (2018). The topological indices
of the Cayley graph of Dihedral group D2n and the Generalized Quaternion group Q2n . Italian
Journal of Pure and Applied Mathematics, 40, pp. 424–433.
[15] Wagner, S. (2007). Extremal trees with respect to Hosoya index and Merrifield-Simmons index.
MATCH Communications in Mathematical and in Computer Chemistry, 57(1), pp. 221–233.
[16] Wiener, H. (1947). Structural determination of paraffin boiling points. Journal of the American
Chemical Society, 69(1), pp. 17–20. doi: 10.1021/ja01193a005.

  • تاریخ دریافت 20 اسفند 1403
  • تاریخ بازنگری 22 شهریور 1404
  • تاریخ پذیرش 29 آبان 1404