Multivariate extended adjacency degree based graphical index: mathematical and chemical significances

Document Type : Research Paper

Authors

1 Deparatment of Mathematics, Jnanabharathi Campus, Bangalore university, Bengaluru-560056

2 Department of Mathematics, Jnanabharathi Campus, Bangalore University, Bengaluru- 560056

Abstract

The Multivariate extended adjacency graphical index, which helps to analyze the molecular structures and predict properties of compounds, is that specific cases for randomly chosen values of the non zero real numbers a, b and c, which are coincide with the vast majority of pre-defined graphical indices being considered. In this paper, we obtained the index value for some specific families of graphs, bounds and characterization in terms of order, size, minimum and maximum degree. Also, we present the chemical applicability for some linear molecular graph of above said graphical indices.

Keywords

Main Subjects


[1] B. Chaluvaraju, and Ameer Basha Shaikh. ”Different Versions of Atom-Bond Connectivity Indices of Some Molecular Structures: Applied for the Treatment and Prevention of COVID-19.” Polycyclic Aromatic Compounds 42(2) (2021), 1–15.
[2] B. Chaluvaraju, H. S. Boregowda and I. N. Cangul, Some Inequalities for the First General Zagreb Index of Graphs and Line Graphs. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. (2020).
[3] K. C. Das, I. Gutman and B. Furtula, On spectral radius and energy of extended adjacency matrix of graphs, Applied Mathematics and Computation, 296 (2017), 116–123.
[4] H.G. Franck and J.W. Stadelhofer, Anthracene-production and uses, Industrial Aromatic Chemistry, (1988), 343–361.
[5] Fu Feng, Bo Deng, and Hongyu Zhang, The Extended Adjacency Indices for Several Types of Graph Operations, International Journal of Systems Science and Applied Mathematics, 8(1) (2023), 7–11.
[6] Gorgan Unger, A Short Note on Benzene and its Properties, Journal of Medicinal and Organic Chemistry, 7(5) (2024), 241–242.
[7] I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin (1986).
[8] I. Gutman, V. R. Kulli, B. Chaluvaraju and H. S. Boregowda, On Banhatti and Zagreb Indices, J. Int. Math. Virtual Inst. 7 (2017), 53–67.
[9] I. Gutman and N. Trinajsti´c, Graph Theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), 535–538.
[10] V. R. Kulli, B. Chaluvaraju, and H. S. Boregowda, ”The product connectivity Banhatti index of a graph, Discussiones Mathematicae Graph Theory, 39(2) (2019): 505–517.
[11] V. R. Kulli, B. Chaluvaraju, and H. S. Boregowda, Some bounds on sum connectivity Banhatti index of graphs, Palestine Journal of Mathematics 8(2) (2019): 355–364.
[12] G. Sabidussi,Graph Multiplication, Mathematische Zeitschrift, 72(1960), 446–457.
[13] Modjtaba Ghorbani, Xueliang Li, Samaneh Zangi, and Najaf Amraei, On the eigenvalue and energy of extended adjacency matrix Applied Mathematics and Computation, 397 (2021), 125939.
[14] F. Harary,Graph theory, Addison-Wesley, Reading Mass., (1969). 
[15] N. Harish, C. Nandheesh Kumar and B Chaluvaraju, An Elliptic-Eccentric Sombor index of a graph and its chemical applicabilities, The Journal of Hyperstructures, 13(2), (2024), 222–236.
[16] Richard Hammack, Wilfried Imrich, and Sandi Klavˇzar, Handbook of Product Graphs, Boca Raton, FL:CRC Press, 2, 2011.
[17] S. T. Hedetniemi, S. M. Hedetniemi, and R. C. Laskar, A Survey of Graph Products, Congressus Numerantium, 47 (1985), 257–272.
[18] W.H. Mills, M. Mills, Describes the first chemical synthesis of pentacene and some of its derivatives, J. Chem. Soc., 101 (1912), 2194.
[19] Sourav Mondal, Nilanjan De, and Anita Pal. ”Topological Indices of Some Chemical Structures Applied for the Treatment of COVID-19 Patients.” Polycyclic Aromatic Compounds (2020), 1–15.
[20] V. S. Shegehalli, and Rachanna Kanabur. Arithmetic-geometric indices of path graph, J. Math. Comput. Sci., 16 (2015), 19–24.
[21] T. Takahashi, T. Takenobu, J. Takeya and Y. Iwasa, Ambipolar Light-Emitting Transistors of a Tetracene Single Crystal, Advanced Functional Materials, 17(10) (2007), 1623-–1628.
[22] R. Todeschini and V. Consonni, Molecular Descriptors for Chemoinformatics, Wiley–VCH, Weinheim (2009).
[23] Trinajstic, Nenad. Chemical graph theory. Routledge, (2018).
[24] Tomlinson, Muriel, An Introduction to the Chemistry of Benzenoid Compounds, Elsevier, (1971), 207. 
[25] US Environmental Protection Agency. Office ofWater Regulations and Standards. Ambient water quality criteria for naphthalene, (1980).
[26] D. Vukiˇcevi´c and B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, Journal of mathematical chemistry, 46 (2009), 1369–1376.
[27] D. Vukiˇcevi´c and M. Gaˇsperov, Bond Additive Modeling - 1. Adriatic indices Croatica chemica acta, 83(3) (2010), 243–260.
[28] D. Vukiˇcevi´c, Bond Additive Modeling - 2. Mathematical properties of max-min rodeg index, Croatica chemica acta, 83(3) (2010), 261–273.
[29] Wilfried Imrich and Sandi Klavˇzar, Cartesian Product of Graphs and Some Applications, Topics in Combinatorics and Graph Theory, ed. Heidelberg: Physica-Verlag, 1990, 179–194.
[30] Yang Bin, Vinayak V. Manjalapur, Sharanu P. Sajjan, Madhura M. Mathai, and Jia-Bao Liu, On extended adjacency index with respect to acyclic, unicyclic and bicyclic graphs, Mathematics, 7(7) (2019), 652.
[31] Yi-Qiu Yang, Lu Xu, and Chang-Yu Hu, Extended adjacency matrix indices and their applications, Journal of Chemical Information and Computer Sciences, 34(5) (1994), 1140–1145.
[32] Z. Wang, Y. Mao, Y. Li, B. Furtula, On relations between Sombor and other degree-based indices, J. Appl. Math. Comput. 68 (2022) 1-17.