Login Register Cart 0
Generic placeholder image

Current Organic Synthesis

Editor-in-Chief

ISSN (Print): 1570-1794
ISSN (Online): 1875-6271

Back
Journal Home About Journal Editorial Board Journal Insight Current Issue Volumes/Issues
Subscribe
Research Article

The Second Omega Index

Author(s): Nurten Urlu Ozalan*, Ahmet Sinan Cevik and Ismail Naci Cangul

Volume 21, Issue 3, 2024

Published on: 14 December, 2023

Page: [286 - 291] Pages: 6

DOI: 10.2174/0115701794250566231115075551

Price: $65

Abstract

Background: The omega index has been recently introduced to identify a variety of topological and combinatorial aspects of a graph with a new viewpoint. As a continuing study of the omega index, by considering the incidence of edges and vertices to the adjacency of the vertices, in this paper, we have introduced the second omega index Ω2 and then computed it over some well-known graph classes.

Methods: Many combinatorial counting methods have been utilized in the proofs. The edge partition is frequently used throughout the work. Naturally, some graph theoretical lemmas are also used.

Results: In particular, trees having the smallest and largest Ω2 have been constructed. Finally, the second omega index of some derived graphs, such as line graphs, subdivision graphs, and vertex-semitotal graphs, has been presented.

Conclusion: Omega invariant has already been explored in many papers. It has been defined in terms of vertex degrees. Vertices correspond to the atoms in a molecule and a calculation, which only depends on the atomic parameters, is not even comparable with a calculation containing both atoms and chemical bonds between them. With this idea in mind, we have evaluated some mathematical properties of the second omega index, which has great potential in chemical applications related to the number of cycles in the molecular graph.

« Previous Next »
Graphical Abstract

[1]
Bondy, J.A.; Murty, U.S.R. Graph Theory; Springer, 2008.
[http://dx.doi.org/10.1007/978-1-84628-970-5]
[2]
Harary, F. Graph Theory; Addison-Wesley: Reading, Mass, 1972.
[3]
Gutman, I.; Trinajstić, N. Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons. Chem. Phys. Lett., 1972, 17(4), 535-538.
[http://dx.doi.org/10.1016/0009-2614(72)85099-1]
[4]
Gutman, I.; Ruscic, B.; Trinajstic, N.; Wilcox, C.F., Jr Graph theory and molecular orbitals, XII. Acyclic polyenes. J. Chem. Phys., 1975, 62(9), 3399-3405.
[http://dx.doi.org/10.1063/1.430994]
[5]
Akgunes, N.; Aydin, B. Introducing new exponential Zagreb indices for graphs. J. Math., 2021, 13.
[http://dx.doi.org/10.1155/2021/6675321]
[6]
Das, K.C.; Xu, K.; Cangul, I.N.; Cevik, A.S.; Graovac, A. On the Harary index of graph operations. J. Inequal. Appl., 2013, 2013(1), 339.
[http://dx.doi.org/10.1186/1029-242X-2013-339]
[7]
Das, K.C.; Çevik, A.S.; Cangul, I.N.; Shang, Y. On Sombor index. Symmetry, 2021, 13(1), 140.
[http://dx.doi.org/10.3390/sym13010140]
[8]
Doslic, T.; Furtula, B.; Graovac, A.; Gutman, I.; Moradi, S.; Yarahmadi, Z. On vertex degree based molecular structure descriptors. MATCH Commun. Math. Comput. Chem, 2011, 66(2), 613-626.
[9]
Gutman, I.; Das, K.C. The first Zagreb index 30 years after. MATCH Commun. Math. Comput. Chem, 2004, 50(1), 83-92.
[10]
Ranjini, P.S.; Lokesha, V.; Cangül, I.N. On the Zagreb indices of the line graphs of the subdivision graphs. Appl. Math. Comput., 2011, 218(3), 699-702.
[http://dx.doi.org/10.1016/j.amc.2011.03.125]
[11]
Delen, S.; Naci Cangul, I. A new graph invariant. Turk. J. Analy. Num. Theory, 2018, 6(1), 30-33.
[http://dx.doi.org/10.12691/tjant-6-1-4]
[12]
Ascioglu, M.; Demirci, M.; Cangul, I.N. Omega invariant of union, join and corona product of two graphs. Adv. Stud. Contemp. Math., 2020, 30(3), 297-306.
[13]
Delen, S.; Togan, M.; Yurttas, A.; Ana, U.; Cangu, I. The effect of edge and vertex deletion on omega invariant. Appl. Ana. Discrete Math., 2020, 14(3), 685-696.
[http://dx.doi.org/10.2298/AADM190219046D]
[14]
Delen, S.; Demirci, M.; Cevik, A.S.; Cangul, I.N. On Omega index and average degree of graphs. J. Math., 2021, 2021, 5.
[http://dx.doi.org/10.1155/2021/5565146]
[15]
Demirci, M.; Delen, S.; Cevik, A.S.; Cangul, I.N. Omega index of line and total Graphs. J. Math., 2021, 2021, 6.
[http://dx.doi.org/10.1155/2021/5552202]
[16]
Gunderson, D.S. Handbook of Mathematical Induction, Theory and Applications; CRC Press, 2014, p. 240.
[http://dx.doi.org/10.1201/b16005]
[17]
Mishra, V.N.; Delen, S.; Cangul, I.N. Degree sequences of join and corona products of graphs. Electron. J. Math. Anal. Appl., 2019, 7(1), 5-13.
[18]
Ranjini, P.S.; Lokesha, V.; Rajan, M.A. On Zagreb indices of the subdivision graphs. Int. J. Math. Sc. Eng. Appl, 2010, 4(4), 221-228.
[19]
Gutman, I.; Yeh, Y. N.; Lee, S. L.; Luo, Y. L. Some recent results in the theory of the Wiener number. Indian J. Chem., 1993, 32, 651-661.
[20]
Nilanjan, D. F-index and coindex of some derived graphs. arXiv, 2016, 2016, 02175.
[21]
Togan, M.; Gunes, A.Y.; Delen, S.; Cangul, I.N. Omega invariant of the line graphs of unicyclic graphs. Montes Taurus J. Pure Appl. Math., 2020, 2(2), 45-48.
[22]
Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica; Addison-Wesley, 1990.

Rights & Permissions Print Cite
© 2024 Bentham Science Publishers | Privacy Policy