Neighborhood First Zagreb Index and Maximal Unicyclic and Bicyclic Graphs

Articles in Press

Document Type : Original paper

Authors

Department of Mathematics , University of Gujrat, Hafiz Hayat Campus, Gujrat, Pakistan

Abstract

The Neighborhood First Zagreb Index NM1 measures the topological properties of a molecular graph. Neighborhood First Zagreb Index NM1 is defined as NM1(G)=∑v∈V(G)(S(v))2, where S(v) used to represent the sum of degrees of vertices adjacent to a vertex v in a graph G.  In this study, we focus on characterizing the graphs with the maximum neighborhood first Zagreb index in the class of unicyclic/bicyclic graphs on n vertices, where n is a fixed integer greater than or equal to 5. Specifically, we are interested in identifying the graphs that have the highest value according to the recently introduced neighborhood first Zagreb index NM1.

Keywords

Main Subjects

References

[1] H. Abdo, D. Dimitrov, T. Réti, and D. Stevanović, Estimation of the spectral radius of graph by the second Zagreb index, MATCH Commun. Math. Comput. Chem. 72 (2014), no. 3, 741–751.
[2] A. Ali, I. Gutman, E. Milovanovćc, and I. Milovanović, Sum of powers of the degrees of graphs: extremal results and bounds, MATCH Commun. Math. Comput. Chem. 80 (2018), no. 1, 5–84.
[3] A. Ali, Z. Raza, and A.A. Bhatti, Bond incident degree (BID) indices of polyomino chains: a unified approach, Appl. Math. Comput. 287 (2016), 28–37.
https://doi.org/10.1016/j.amc.2016.04.012
[4] A. Ali, L. Zhong, and I. Gutman, Harmonic index and its generalizations: extremal results and bounds, MATCH Commun. Math. Comput. Chem. 81 (2019), no. 2, 249–311.
[5] D. Bonchev and L.B. Kier, Topological atomic indices and the electronic charges in alkanes, J. Math. Chem. 9 (1992), no. 1, 75–85.
https://doi.org/10.1007/BF01172931
[6] B. Borovićanin, K.C. Das, B. Furtula, and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78 (2017), 17–100.
[7] K.C. Das, Maximizing the sum of the squares of the degrees of a graph, Discrete Math. 285 (2004), no. 1-3, 57–66.
https://doi.org/10.1016/j.disc.2004.04.007
[8] K.C. Das, K. Xu, and J. Nam, Zagreb indices of graphs, Front. Math. China 10 (2015), no. 3, 567–582.
https://doi.org/10.1007/s11464-015-0431-9
[9] T. Došlić and T. Réti, Novel degree-based molecular descriptors with increased discriminating power, Acta Polytech. Hung. 9 (2012), no. 4, 17–30.
[10] S. Fajtlowicz, On conjectures of Graffiti-II, Congr. Numer 60 (1987), 187–97.
[11] M. Ghorbani and M.A. Hosseinzadeh, Computing ABC4 index of nanostar dendrimers, Optoelectron. Adv. Mat. 4 (2010), 1419–1422.
[12] C. Godsil and G. Royle, Algebraic Graph Theory, Springer New York, 2001.
[13] A. Graovac, M. Ghorbani, and M.A. Hosseinzadeh, Computing fifth geometric-arithmetic index for nanostar dendrimers, J. Disc. Math. Appl. 1 (2011), no. 1-2, 33–42.
https://doi.org/10.22061/jmns.2011.461
[14] J.L. Gross and J. Yellen, Graph Theory, CRC Press, Boca Raton, Florida, 2000.
[15] I. Gutman, Degree-based topological indices, Croat. Chem. Acta, CCA) 86 (2013), no. 4, 351–361.
http://dx.doi.org/10.5562/cca2294
[16] I. Gutman and K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), 83–92.
[17] I. Gutman, E. Milovanović, and I. Milovanović, Beyond the Zagreb indices, AKCE Int. J. Graphs Comb. 17 (2020), no. 1, 74–85.
https://doi.org/10.1016/j.akcej.2018.05.002
[18] I. Gutman and J. Tošović, Testing the quality of molecular structure descriptors. Vertex-degree-based topological indices, J. Serb. Chem. Soc. 78 (2013), no. 6, 805–810.
[19] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total ϕ-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538.
https://doi.org/10.1016/0009-2614(72)85099-1
[20] I. Gutman, N. Trinajstić, and C.F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys. 62 (1975), no. 9, 3399–3405.
https://doi.org/10.1063/1.430994
[21] B. Liu and Z. You, A survey on comparing Zagreb indices, MATCH Commun. Math. Comput. Chem. 65 (2011), no. 3, 581–593.
[22] I.Z. Milovanovic, V.M. Ćirić, I. Milovanović, and E. Milovanović, On some spectral, vertex and edge degree–based graph invariants, MATCH Commun. Math. Comput. Chem 77 (2017), no. 1, 177–188.
[23] S. Mondal, N. De, and A. Pal, On neighborhood Zagreb index of product graphs, J. Mol. Struct. 1223 (2021), Article ID: 129210.
https://doi.org/10.1016/j.molstruc.2020.129210
[24] S. Nikolić, G. Kovačević, A. Miličević, and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003), no. 2, 113–124.
[25] T. Réti, A. Ali, P. Varga, and E. Bitay, Some properties of the neighborhood first Zagreb index, Discrete Math. Lett. 2 (2019), 10–17.
[26] T. Réti, I. Gutman, and D. Vukičević, On Zagreb indices of pseudo-regular graphs, J. Discrete Math. Appl. 1 (2011), no. 1-2, 1–12.
https://doi.org/10.22061/jmns.2011.458
[27] D. Stevanović, Spectral Radius of Graphs, Academic Press, Amsterdam, 2015.
[28] A.A. Toropov, A.P. Toropova, T.T. Ismailov, N.L. Voropaeva, and I.N. Ruban, Extended molecular connectivity: Prediction of boiling points of alkanes, J. Struct. Chem. 38 (1997), no. 6, 965–969.
https://doi.org/10.1007/BF02763818
[29] S. Wang and B. Zhou, On the first extended zeroth-order connectivity index of trees, Iran. J. Sci. 38 (2014), no. 3, 213–219.
[30] 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), no. 2, 109–128.
https://doi.org/10.22052/ijmc.2022.242939.1571
[31] A. Yu, M. Lu, and F. Tian, On the spectral radius of graphs, Linear Algebra Appl. 387 (2004), 41–49.
https://doi.org/10.1016/j.laa.2004.01.020
[32] B. Zhou and N. Trinajstić, On extended connectivity indices, J. Math. Chem. 46 (2009), no. 4, 1172–1180.
https://doi.org/10.1007/s10910-008-9500-6
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 06 September 2024

Files

Share

How to cite

Statistics