THE FIRST AND SECOND ZAGREB INDICES TO MOLECULAR GRAPHS OF ALKANES, C_n H_((2n+2) )
Abstract
For a simple graph G=(V(G) ,E(G)) , let d(u), d(v) be the degree of the vertices u, and v
in G. The first and second Zagreb indices of G are defined as
M_1 (G) = ∑_(e=uv∈E(G))▒〖(d(u)+d(v))〗 and M_2 (G) = ∑_(e=uu∈E(G))▒〖d(u)d(v)〗, respectively.
The first, generalized, and the second Multiplicative Zagreb indices of simple graph G are defined as
〖PM〗_1 (G)= ∏_(e=uv∈E(G))▒(d(u)+d(v)) and 〖PM〗_2 (G)= ∏_(e=uu∈E(G))▒(d(u)d(v))
respectively. The multiplicative Zagreb indices have been the focus of considerable research in computational chemistry dating back to Narumi and Katayama in 1980s.
The general class of molecules known as the alkanes, or paraffins, is given by chemical formula C_n H_((2n+2) ) and it is natural to ask how many different molecules, number of atoms are there a with this formula. We derive a simple formula to find the first and second Zagreb indices for any positive integer n , show formulate a python approach to find a vertex distance matrix from a vertex incidence matrix, formulate Zagreb indices by (d(e)), the degree set of adjacent edges, and prove
M_1 (G)= ∑_(e ∈E(G))▒〖d(e)〗 + ∑_(u ∈V(G))▒〖d(u)〗,
∑_(e=uv∈E(G))▒〖(d(u))^2+(d(v))^2 〗 = ∑_(u∈V(G))▒(d(u))^3
and the First and the second multiplicative Zagreb indices of a graph G
〖PM〗_1 (G)=∏_(e=uv∈E(G))▒(d(e)+2) and 〖PM〗_2 (G)=∏_(u∈V(G))▒(d(u))^((d(u)) )
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References
Zagreb indices and walk counts, MATCH Commun. Math. Comput. Chem., 54 (2005), 163-176.
[2] B. Basavana Goud, S. Patil, A note on hyper–Zagreb index of graph operations, Iran. J. Math. Chem. 7
(2016) 89–92.
[3] B. Borovi´canin, K. C. Das, B. Furtula, I. Gutman, Zagreb indices: Bounds and extremal graphs, in: I.
Gutman, B. Furtula, K. C. Das, E. Milovanovi´c, I. Milovanovi´c (Eds.), Bounds in Chemical Graph Theory –
Basics, Univ. Kragujevac, Kragujevac, 2017, pp. 67–153.
[4] B. Borovi´canin, K. C. Das, B. Furtula, I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math.
Comput. Chem. 78 (2017) 17–100.
[5] S¸. B. Bozkurt, A. D. G¨ung¨or, I. Gutman, Randi´c spectral radius and Randi´c energy, MATCH Commun.
Math. Comput. Chem. 64 (2010) 321–334.
[6] S¸. B. Bozkurt, A. D. G¨ung¨or, I. Gutman, A. S C¸ evik, Randi´c matrix and Randi´c energy, MATCH
Commun. Math. Comput. Chem. 64 (2010) 239–250.
[7] B. Zhou, Z. Du, Some lower bounds for Estrada index, Iran. J. Math. Chem. 1 (2010)
[8] B. Zhou and I. Gutman, Relations between Wiener, hyper-Wiener and Zagreb indices, Chemical Physics
Letters, 394 (2004), 93-95.
[9] B. Zhou and N. Trinajsti´c, On a novel connectivity index, J. of Mathematical Chemistry, 46 (2009) 1252-
1270.
[10] B. Zhou and N. Trinajsti´c, On general sum-connectivity index, Journal of Mathematical Chemistry, 47
(2010), 210-218
[11] D. Cvetkovi´c, P. Rowlinson, S. Simi´c, An Introduction to the Theory of Graph Spectra, Cambridge Univ.
Press, Cambridge, 2010. -385-
[12] Z. Cvetkovski, Inequalities, Theorems, Techniques and Selected Problems, Springer, Berlin, 2012.
[13] K. C. Das, S. Sorgun, K. Xu, On Randi´c energy of graphs, in: I. Gutman, X. Li (Eds.), Graph Energies –
Theory and Applications, Univ. Kragujevac, Kragujevac, 2016, pp. 111–122.
[14] H. Deng, S. Radenkovi´c, I. Gutman, The Estrada Index, in: D. Cvetkovi´c, I. Gutman (Eds.), Applications of
Graph Spectra, Math. Inst., Belgrade, 2009, pp. 123–140.
[15] E. Estrada, The ABC matrix, J. Math. Chem. 55 (2017) 1021–1033. [16] I. Gutman, Degree–based
topological indices, Croat. Chem. Acta 86 (2013) 351–361.
[17] I. Gutman, On hyper–Zagreb index and coindex, Bull. Acad. Serbe Sci. Arts (Cl. Math. Natur.), in press.
[18] I. Gutman, K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50
(2004) 83–92.
[19] I. Gutman, B. Ruˇsˇci´c, N. Trinajsti´c, C. F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic
polyenes, J. Chem. Phys. 62 (1975) 3399–3405.
[20] I. Gutman, N. Trinajsti´c, Graph theory and molecular orbitals. Total π-electron energy of alternant
hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538.
[21] S. M. Hosamani, B. B. Kulkarni, R. G. Boli, V. M. Gadag, QSPR analysis of certain graph theoretical
matrices and their corresponding energy, Appl. Math. Nonlin. Sci. 2 (2017) 131–150.
[22] S. M. Hosamani, H. S. Ramane, On degree sum energy of a graph, Europ. J. Pure Appl. Math. 9 (2016)
340–345.
[23] N. Jafari Rad, A. Jahanbani, I. Gutman, Second Zagreb energy and second Zagreb Estrada index of
graphs, forthcoming.
[24] X. Li, Y. Shi, I. Gutman Graph Energy, Springer, New York, 2012.
[25] S. Nikoli´c, G. Kovaˇcevi´c, A. Miliˇcevi´c, N. Trinajsti´c, The Zagreb indices 30 years after, Croat. Chem.
Acta 76 (2003) 113–124.
[26] K. Pattabiraman, M. Vijayaragavan, Hyper Zagreb indices and its coindices of graphs, Bull. Int. Math. Virt.
Inst. 7 (2017) 31–41.
[27] H. S. Ramane, D. S. Revankar, J. B. Patil, Bounds for the degree sum eigenvalues and degree sum energy
of a graph, Int. J. Pure Appl. Math. Sci. 6 (2013) 161-167.
[28] J. M. Rodr´ıguez, J. M. Sigarreta, Spectral properties of geometric–arithmetic index, Appl. Math. Comput.
277 (2016) 142–153.
[29] G. H. Shirdel, H. Rezapour, A. M. Sayadi, The hyper–Zagreb index of graph operations, Iran. J. Math.
Chem. 4 (2013) 213–220.
[30]. R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley VCH, Weinheim, 2000.
[31] Robin J. Wilson, “Introduction to Graph Theory”, England, Addison Wesley Longman Limited, 1998.
Fourth edition, ISBN 0-582-24993-7.
[31] Stephen Wagner and Hua Wand. “Introduction to Chemical Graph Theory”, New York, Chapman and
Hall/CRC, 2018. First edition, ISBN 978113832508.
[32] Zhou B, Xing R. On Atomic Bond Connectivity Index Z Naturforsch. 2011; 66a:61-66
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