On the Multiplicative‎ ‎Reformulated‎ ‎First‎ ‎Zagreb Index of n-Vertex‎ ‎Trees ‎ ‎with Respect to Matching Number

Document Type : Research Paper

Authors

Department of Mathematics‎, ‎Faculty of Science‎, University of Gujrat‎, ‎Gujrat‎, ‎Pakistan

Abstract

‎The multiplicative first Zagreb index is the product of the square of the degree of vertices in a graph $\mathbb{G}$‎. ‎The multiplicative reformulated first Zagreb index is defined as $\prod_{1,e}(\mathbb{G})= \prod_{x_{1}x_{2}\in E(\mathbb{G})}(d_{\mathbb{G}}(x_{1})+d_{\mathbb{G}}(x_{1})-2)^{2}$‎, ‎where $E(\mathbb{G})$ is the edge set of a graph $\mathbb{G}$ and $d_{\mathbb{G}}(x_{1})$ is the degree of a vertex $x_{1}$ in a graph $\mathbb{G}$‎. ‎In this paper‎, ‎we characterize the minimum and maximum trees and unicyclic graphs with respect to matching and perfect matching using this graph invariant $\prod_{1,e}(\mathbb{G})$ among the collection of all $n$-vertex graphs‎.

Keywords

Main Subjects

References

[1] C. Godsil and G. F. Royle, Algebraic Graph Theory, Springer Science & Business Media, 2001.
[2] B. Bollobas, Extremal Graph Theory, Courier Corporation, 2004.
[3] I. Gutman and N. Trinajstic, Graph theory and molecular orbitals. Total $\varphi$-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538, https://doi.org/10.1016/0009-2614(72)85099-1.
[4] G. H. Shirdel, H. Rezapour and A. M. Sayadi, The hyper-Zagreb index of graph operations, Iranian J. Math. Chem. 4 (2013) 213–220, https://doi.org/10.22052/IJMC.2013.5294.
[5] B. Furtula, I. Gutman and S. Ediz, On difference of Zagreb indices, Discrete Appl. Math. 178 (2014) 83–88, https://doi.org/10.1016/j.dam.2014.06.011.
[6] S. Ediz, On the reduced first Zagreb index of graphs, Pac. J. Appl. Math. 8 (2016) 99–102.
[7] K. Xu, K. Tang, H. Liu and J. Wang, The Zagreb indices of bipartite graphs with more edges, J. Appl. Math. Inform. 33 (2015) 365–377.
[8] A. Milicevic', S. Nikolic' and N. Trinajstic', On reformulated Zagreb indices, Mol. Divers 8 (2004) 393–399, https://doi.org/10.1023/B:MODI.0000047504.14261.2a.
[9] K. C. Das, On comparing Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 63 (2010) 433–440.
[10] K. Xu and H. Hua, A unified approach to extremal multiplicative Zagreb indices for trees, unicyclic and bicyclic graphs, MATCH Commun. Math. Comput. Chem. 68 (2012) 241–256.
[11] Z. Yan, H. Liu and H. Liu, Sharp bounds for the second Zagreb index of unicyclic graphs, J. Math. Chem. 42 (2007) 565–574, https://doi.org/10.1007/s10910-006-9132-7.
[12] A. Chang and F. Tian, On the spectral radius of unicyclic graphs with perfect matchings, Linear Algebra Appl. 370 (2003) 237–250, https://doi.org/10.1016/S0024-3795(03)00394-X.
[13] X. Li and J. Wang, On the ABC spectra radius of unicyclic graphs, Linear Algebra Appl. 596 (2020) 71–81, https://doi.org/10.1016/j.laa.2020.03.007.
[14] H. Liu, M. Lu and F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, Linear Algebra Appl. 420 (2007) 449–457, https://doi.org/10.1016/j.laa.2006.08.002.
[15] J. B. Lv, J. Li and W. C. Shiu, The harmonic index of unicyclic graphs with given matching number, Kragujevac J. Math. 38 (2014) 173–183.
[16] L. Zhong, The harmonic index for unicyclic and bicyclic graphs with given matching number, Miskolc Math. Notes 16 (2015) 587–605, https://doi.org/10.18514/MMN.2015.1033.
[17] T. Zhou, Z. Lin and L. Miao, The extremal Sombor index of trees and unicyclic graphs with given matching number, J. Discrete Math. Sci. Cryptogr. (2022) 1–12, https://doi.org/10.1080/09720529.2021.2015090.
[18] A. Alidadi, A. Parsian and H. Arianpoor, The minimum Sombor index for unicyclic graphs with fixed diameter, MATCH Commun. Math. Comput. Chem. 88 (2022) 561–572, https://doi.org/10.46793/match.88-3.561A.
[19] S. Yousaf, A. A. Bhatti and A. Ali, On the minimum variable connectivity index of unicyclic graphs with a given order, Discrete Dyn. Nat. Soc. 2020 (2020) Article ID 1217567, https://doi.org/10.1155/2020/1217567.
[20] S. Adeel and A. A. Bhatti, On the extremal total irregularity index of nvertex trees with fixed maximum degree, Commun. Comb. Optim. 6 (2021) 113–121, https://doi.org/10.22049/CCO.2020.26965.1168.
[21] S. Yousaf and A. A. Bhatti, Maximum variable connectivity index of n-vertex trees, Iranian J. Math. Chem. 13 (2022) 33–44, https://doi.org/10.22052/IJMC.2022.243077.1584.
[22] S. Yousaf, A. A. Bhatti and A. Ali, On total irregularity index of trees with given number of segments or branching vertices, Chaos Solitons Fractals 157 (2022) p. 111925, https://doi.org/10.1016/j.chaos.2022.111925.
[23] S. Yousaf and A. A. Bhatti, Maximum total irregularity index of some families of graph with maximum degree n 􀀀 1, Asian-Eur. J. Math. 15 (2022) p. 2250069, https://doi.org/10.1142/S1793557122500693.
[24] S. Yousaf and A. A. Bhatti, On the minimal unicyclic and bicyclic graphs with respect to the neighborhood first Zagreb index, Iranian J. Math. Chem. 13 (2022) 109–128, https://doi.org/10.22052/IJMC.2022.242939.1571.
[25] I. Gutman, Multiplicative Zagreb indices of trees, Bull. Soc. Math. Banja Luka. 18 (2011) 17–23.
[26] A. Ali, A. Nadeem, Z. Raza, W. W. Mohammed and E. M. Elsayed, On the reformulated multiplicative first Zagreb index of trees and Unicyclic graphs, Discrete Dyn. Nat. Soc. 2021 (2021) Article ID 3324357, https://doi.org/10.1155/2021/3324357.
[27] M. Aruvi, The multiplicative reformulated first Zagreb index of some graph operations, Malaya J. Mat. 8 (2020) 1189–1195, https://doi.org/10.26637/MJM0803/0079.
Iranian Journal of Mathematical Chemistry
Volume 15, Issue 3
September 2024
Pages 203-225

Files

Share

How to cite

Statistics