Elmas Çizgelerin Bazı Ters Derece Tabanlı Topolojik İndekslerinin Hesaplanması

Öz

Fen bilimleri, mühendislik bilimleri ve sosyal bilimler gibi birçok farklı bilim dalında çizge teori çalışılmaktadır. Özellikle matematik, kimya, bilgi ve ağ analizi alanlarda çizge teorinin uygulamaları görülmektedir. Son zamanlarda matematiksel kimya alanında bir molekülün kimyasal analizi çizge teorik parametreleri ile hesaplanabilmektedir. Bu parametreler topolojiksel çizge indeksleri olarak tanımlanır. Bu indekslerin farklı özelliklere göre tanımları vardır, örneğin dereceye bağlı topolojiksel indeksler ve dışmerkezliğe dayalı topolojiksel indeksler literatürde çalışılmaktadır. Son zamanlarda ters topolojiksel indeksler tanımlanmıştır. Bu çalışmada, elmas çizgelerin bazı önemli ters derece tabanlı topolojiksel indeks değerleri incelenmiş ve bunlar için tam sonuçlar elde edilmiştir.

Anahtar Kelimeler

• Matematiksel kimya
• Kimyasal çizge teori
• Derece tabanlı topolojik indeksler
• Ters topolojik indeksler
• Elmas çizgeler

Computing Some Reverse Degree Based Topological Indices of Diamond Graphs

Öz

Graph theory is studied in many different branches of science such as science, engineering and social sciences. Applications of graph theory are seen especially in mathematics, chemistry, information and network analysis. Recently, in mathematical chemistry, chemical analysis of a molecule can be calculated with graph theoretical parameters. These parameters are defined as topological graph indices. These indices have definitions according to different properties, for example, degree-based topological indices and eccentricity-based topological indices are studied in the literature. Recently, reverse topological indices have been defined. In this study, some important reverse topological indices of diamond graphs have been studied and exact results have been obtained for them.

Anahtar Kelimeler

• Mathematical chemistry
• Chemical graph theory
• Degree-based topological indices
• Reverse topological indices
• Diamond graphs

Kaynakça

1. Ahmad A., Koam AN., Azeem M. Reverse-degree-based topological indices of fullerene cage networks. Molecular Physics 2023; 121(14): e2212533.
2. Andova V., Dimitrov D., Fink J., Skrekovski R. Bounds on Gutman Index. MATCH Communications in Mathematical and in Computer Chemistry 2012; 67: 515-524.
3. Aslan E. The edge eccentric connectivity index of armchair polyhex nanotubes. Journal of Comp. and Theor. Nano. 2015a; 12(11): 4455-4458.
4. Aslan E. On the fourth atom-bond connectivity index of nanocones. Optoelectronics and Advanced Materials-Rapid Communications 2015b; 9(3-4): 525-527.
5. Aslan E., Kürkçü ÖK. Edge harmonic index of carbon nanocones and an algorithm, Bulgarian Chemical Communications 2018; 50(3): 478-483.
6. Balaban AT., Devillers J. Topological indices and related descriptors in QSAR and QSPAR, CRC Press 2014. Chartrand G., Lesniak L. Graphs and digraphs. California: Chapman and Hall 1996.
7. Çolakoğlu Havare, Ö. Reformulated Zagreb indices of some cycle-related graphs and linear [n]-phenylenes. Osmaniye Korkut Ata Üniversitesi Fen Bilimleri Enstitüsü Dergisi 2024; 7(1): 33-45.
8. Dobrynin AA., Entringer R., Gutman I. Wiener index of trees: theory and applications. Acta Applicandae Mathematicae 2001; 66: 211–249.

9. Ediz S., Cancan M. Reverse Zagreb indices of cartesian product of graphs. International Journal of Mathematics and Computer Science 2016; 11(1): 51–58.
10. Estrada E., Torres L., Rodriguez L., Gutman I. An atom-bond connectivity index: modelling the enthalpy of formation of alkanes. Ind. J. Chem. 1998; 37A: 849–855.
11. Gallian JA. Graph labeling. Electronic Journal of Combinatorics 2014; 17: (Dynamic Survey #DS6).
12. Gowtham KJ., Husin MN. A study of families of bistar and corona product of graph: Reverse topological indices. Malaysian Journal of Mathematical Sciences 2023; 17(4): 575-586.
13. Gowtham KJ. A study of reverse topological indices and their importance in chemical sciences. Applied Mathematics E-Notes 2023; 23: 175–186. Graovac A., Ghorbani M., Hosseinzadeh MA. Computing fifth geometric-arithmetic index for nanostar dendrimers, J. Math. Nanoscience 2011; 1(1): 33–42.
14. Gupta S., Singh M., Madan AK. Connective eccentricity index: a novel topological descriptor for predicting biological activity. Journal of Molecular Graphics and Modelling 2000; 18(1): 18–25.
15. Gutman I., Trinajstic N. Graph theory and molecular orbitals. III. Total j-electron energy of alternant hydrocarbons. Chemical Physics Letters 1972; 17: 535–538.
16. Gutman I., Ruscic B., Trinajstic N. Wilcox CF. Graph theory and molecular orbitals. XII. Acyclic polyenes. Journal of Chemical Physics 1975; 62: 3399–3405.
17. Harary F., Buckley F. Distance in graphs. Addison-Wesley Publishing Company 1989.
18. Hinding N., Firmayasari D., Basir H., Bača M., Feňovčíková AS. On irregularity strength of diamond network. AKCE International Journal of Graphs and Combinatorics 2018; 15(3): 291-297.
19. Jahanbani A., Sheikholeslam S. The topological indices and some Hamiltonian properties of graphs. Applied Mathematics E-Notes 2023; 23: 260-264.
20. Koam AN., Ansari MA., Haider A., Ahmad A., Azeem M. Topological properties of reverse-degree-based indices for sodalite materials network. Arabian Journal of Chemistry 2022; 15(10): 104160.
21. Kulli VR. On the sum connectivity reverse index of oxide and honeycomb networks. Journal of Computer and Mathematical Sciences 2017a; 9(8): 408–413.
22. Kulli VR. Geometric-arithmetic reverse and sum connectivity reverse indices of silicate and hexagonal networks. International Journal of Current Research in Science and Technology 2017b; 3(10): 29–33.
23. Kulli VR. Reverse Zagreb and reverse hyper-Zagreb indices and their polynomials of rhombus silicate networks. Annals of Pure and Applied Mathematics 2018a; 16(1): 47–51.
24. Kulli VR. Computing two arithmetic-geometric reverse indices of certain networks. International Research Journal of Pure Algebra 2018b; 8(8): 43–49.
25. Li R. Bounds of the forgotten topological index and some Hamiltonian properties of graphs. Electronic Journal of Mathematics 2024; 8: 48–53.
26. Li, R. The General Inverse Degree and Some Hamiltonian Properties of Graphs. Journal of Combinatorial Mathematics and Combinatorial Computing 2025; 124: 87-98.
27. Ökten Turaci M. On vertex and edge eccentricity-based topological indices of a certain chemical graph that represents bidentate ligands. Journal of Molecular Structure 2020; 1207: 127766.
28. Ökten Turaci M. The values of eccentricity-based topological indices of diamond graphs. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 2018; 22(Özel Sayı): 285-289.
29. Randic M. On characterization of molecular branching. J. Am. Chem. Soc. 1975; 97: 6609–6615.
30. Shulhany MA., Salman ANM. Bilangan Terhubung Pelangi Graf Berlian. Prosiding Seminar Nasional Matematika dan Pendidikan Matematika 2015; UMS (2015): 916–924.
31. Swamy NN., Manohar T., Sooryanarayana B., Gutman I. Reverse sombor index. Bulletin of International Mathematical Virtual Institute 2022; 12(2): 267–272.
32. Todeschini R., Consonni V. Handbook of molecular descriptors. Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim 2000.
33. Turacı T., Ökten M. The edge eccentric connectivity index of hexagonal cactus chains. Journal of Computational and Theoretical Nanoscience 2015; 12(10): 3977-3980.
34. Vukicevic D., Fortula B. Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges. J. Math. Chem. 2009; 46: 1369–1376.
35. Vukicevic D., Graovac A. Note on the comparison of the first and second normalized Zagreb eccentricity indices. Acta Chimica Slovenica 2010; 57: 524–528.
36. Wiener H. Structrual determination of paraffin boiling points. Journal of the American Chemical Society 1947; 69(1): 17-20.