### Some Algebraic Polynomials and Topological Indices of Octagonal Network

#### Abstract

M-polynomial of different molecular structures helps to calculate many topological indices. A topological index of graph \(G\) is a numerical parameter related to \(G\) which characterizes its molecular topology and is usually graph invariant. In the field of *quantitative structure-activity* (QSAR), *quantitative structure-activity structure-property* (QSPR) research, theoretical properties of the chemical compounds and their molecular topological indices such as the Zagreb indices, Randic index, Symmetric division index, Harmonic index, Inverse sum index, Augmented Zagreb index, multiple Zagreb indices etc. are correlated. In this report, we compute closed forms of M-polynomial, first Zagreb polynomial and second Zagreb polynomial of octagonal network. From the M-polynomial we recover some degree-based topological indices for octagonal network.

#### Keywords

#### Full Text:

PDF#### References

Y.C. Kwun, A. Ali, W. Nazeer, M. Ahmad Chaudhary and S. M. Kang, M-polynomials and degreebased topological indices of triangular, hourglass, and Jagged-rectangle Benzenoid systems, Journal of Chemistry 2018 (2018), Article ID 8213950, 8 pages DOI: 10.1155/2018/8213950.

C. Amic, D. Beslo, B. Lucic, S. Nikolic and N. Trinajsti´c, The vertex-connectivity index revisited, J. Chem. Inf Comput. Sci. 38 (1998), 819 – 822, DOI: 10.1021/ci980039b.

B. Bollobas and P. Erdös, Graphs of extremal weights, Ars Combin. 50 (1998), 225 – 233.

H. Deng, G. Huang and X. Jiang, A unified linear-programming modeling of some topological indices, J. Comb. Opt. 30(3) (2015), 826 – 837, DOI: 10.1007/s10878-013-9672-2.

H. Deng, J. Yang anf F. Xia, A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes, Comp. Math. Appl. 61(2011), 3017 – 3023, DOI: 10.1016/j.camwa.2011.03.089.

E. Deutsch and S. Klavzar, M-polynomial and degree-based topological indices, Iranian Journal of Mathematical Chemistry 6(2) (2015), 93 – 102, DOI: 10.22052/ijmc.2015.10106.

B. Furtula, A. Graovac and D. Vukicevic, Augmented Zagreb index, J. Math. Chem. 48 (2010), 370 – 380, DOI: 10.1007/s10910-010-9677-3.

M. Ghorbani and N. Azimi, Note on multiple Zagreb indices, Iran. J. Math. Chem. 3(2) (2012), 137 – 143, DOI: 10.22052/ijmc.2012.5233.

I. Gutman, Degree-based topological indices, Croat. Chem. Acta 86 (2013), 351 – 361, DOI: 10.5562/cca229.

I. Gutman, Molecular graphs with minimal and maximal Randic indices, Croatica Chem. Acta 75 (2002), 357 – 369.

I. Gutman and K.C. Das, The first Zagreb indices 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), 83 – 92.

I. Gutman and N. Trinajstic, Graph theory and molecular orbitals total '-electron energy of alternanthydrocarbons, Chem. Phys. Lett. 17 (1972), 535–538, DOI: 10.1016/0009-2614(72)85099-1.

I. Gutman, Some properties of the Wiener polynomials, Graph Theory Notes 25 (1993), 13 – 18.

Y. Huang, B. Liu and L. Gan, Augmented Zagreb index of connected graphs, MATCH Commun. Math. Comput. Chem. 67 (2012), 483 – 494.

S. Kang, M. Munir, A. Nizami, Z. Shahzadi and W. Nazeer, Some Topological Invariants of the Möbius Ladder, Preprints 2016 (2016), 2016110040, DOI: 10.20944/preprints201611.0040.v1.

L.B. Kier and L.H. Hall, Molecular Connectivity in Structure-Activity Analysis, John Wiley & Sons, New York, USA (1986), DOI: 10.1002/jps.2600760325.

S. Klav˘zar and I. Gutman, A comparison of the Schultz molecular topological index with the Wiener index, J. Chem. Inf. Comput. Sci. 36(1996), 1001 – 1003, DOI: 10.1021/ci9603689.

X. Li and I. Gutman, Mathematical Aspects of Randic-Type Molecular Structure Descriptors; Mathematical Chemistry Monographs, No. 1; University of Kragujevac, Kragujevac, Serbia (2006).

X. Li and Y. Shi, A survey on the Randic index, MATCH Commun. Math. Comput. Chem. 59(2008), 127 – 156.

M. Munir, W. Nazeer, S. Rafique and S. Kang, M-polynomial and degree-based topological indices of polyhex nanotubes, Symmetry 8(12) (2016), 149, DOI: 10.3390/sym8120149.

M. Munir, W. Nazeer, S. Rafique, A Nizami and S.M. Kang, Some computational aspects of triangular boron nanotubes, Preprints 2016 (2016), 2016110041 DOI: 10.20944/preprints201611.0041.v1.

M. Munir, W. Nazeer, S. Rafique, A.R. Nizami and S.M. Kang, M-polynomial and degree-based topological indices of Titania Nanotubes, Symmetry 8 (2016), 117, DOI: 10.3390/sym8110117.

H. M. ur Rehman, R. Sardar and A. Raza, Computing topological indices of hex board and its line graph, Open J. Math. Sci. 1 (2017), 62 – 71, DOI: 10.30538/oms2017.0007.

G. Rucker and C. Rucker, On topological indices, boiling points, and cycloalkanes, J. Chem. Inf. Comput. Sci. 39(1999), 788, DOI: 10.1021/ci9900175.

M. S. Sardar, S. Zafar and M. R. Farahani, The generalized Zagreb index of Capra-designed planar benzenoid series Cak(C6), Open J. Math. Sci. 1(2017), 44 – 51.

G.H. Shirdel, H.R. Pour and A.M. Sayadi, The hyper-Zagreb index of graph operations, Iran J. Math. Chem. 4(2) (2013), 213 – 220, DOI: 10.22052/ijmc.2013.5294.

S. M. Kang, M. A. Zahid, W. Nazeer and W. Gao, Calculating the degree-based topological indices of dendrimers, Open Chemistry 16(1) (2018), 681 – 688, DOI: 10.1515/chem-2018-0071.

D. Vukicevic, On the edge degrees of trees, Glas. Mat. Ser. III 44(4) (2009), 259 – 266.

D. W. West, An Introduction to Graph Theory, Prentice-Hall (1996).

DOI: http://dx.doi.org/10.26713%2Fjims.v11i3-4.600

eISSN 0975-5748; pISSN 0974-875X