ZAGREB INDICES AND MULTIPLICATIVE ZAGREB INDICES OF DOUBLE GRAPHS OF SUBDIVISION GRAPHS

Yıl 2019, Cilt: 9 Sayı: 2, 404 - 412, 01.06.2019

M. Togan A. Yurttas A. S. Cevik I. N. Cangul

Öz

Let G be a simple graph. The subdivision graph and the double graph are the graphs obtained from a given graph G which have several properties related to the properties of G. In this paper, the rst and second Zagreb and multiplicative Zagreb indices of double graphs, subdivision graphs, double graphs of the subdivision graphs and subdivision graphs of the double graphs of G are obtained. In particular, these numbers are calculated for the frequently used null, path, cycle, star, complete, complete bipartite or tadpole graph.

Anahtar Kelimeler

Zagreb indices, multiplicative Zagreb indices, double graphs, subdivision graphs.

Kaynakça

• [1] A. Ali, Tetracyclic graphs with maximum second Zagreb index: a simple approach, Asian-European J. Math., DOI:10.1142/S1793557118500626, to appear.
• [2] A. R. Ashrafi, T. Doˇsli´c, A. Hamzeh, The Zagreb coindices of graph operations, Discrete Appl. Math., 158 (2010), 1571-1578.
• [3] B. Borovianin, K. C. Das, B. Furtula, I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78 (2017) 17-100.
• [4] K. C. Das, N. Akgunes, M. Togan, A. Yurttas, I. N. Cangul, A. S. Cevik, On the first Zagreb index and multiplicative Zagreb coindices of graphs, Analele Stiintifice ale Universitatii Ovidius Constanta, 24 (1) (2016), 153-176 DOI: 10.1515/auom-2016-0008.
• [5] K. C. Das, N. Trinajsti´c, Relationship between the eccentric connectivity index and Zagreb indices, Comp. Math. Appl., 62 (4) (2011), 1758-1764.
• [6] K. Ch. Das, A. Yurttas, M. Togan, I. N. Cangul, A. S. Cevik, The multiplicative Zagreb indices of graph operations, JIA Journal of Inequalities and Applications, 90 (2013).
• [7] M. Eliasi, A. Iranmanesh, I. Gutman, Multiplicative versions of first Zagreb index, MATCH Commun. Math. Comput. Chem., 68 (2012), 217-230.
• [8] I. Gutman, Multiplicative Zagreb indices of trees, Bulletin of Society of Mathematicians Banja Luka, 18 (2011), 17-23.
• [9] I. Gutman, K. C. Das, The First Zagreb index 30 years after, MATHCH Commun. Math. Comput. Chem., 50 (2004), 83-92.
• [10] 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.
• [11] I. Gutman, N. Trinajstic, Graph theory and molecular orbitals. III. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett., 17 (1972), 535-538.
• [12] S. M. Hosamani, V. Lokesha, I. N. Cangul, K. M. Devendraiah, On Certain Topological Indices of the Derived Graphs of Subdivision Graphs, Turkic World of Mathematical Sciences, Journal of Applied Engineering Mathematics, 6 (2) (2016), 324-332.
• [13] G. Indulal, A. Vijayakumar, On a Pair of Equienergetic Graphs, MATCH Commun. Math. Comput. Chem., 55 (2006) 83-90.
• [14] A. Ili´c, D. Stevanovi´c, On comparing Zagreb indices, MATCH Commun. Math. Comput. Chem. 62 (2009) 681687.
• [15] J. B. Liu, C.Wang, S.Wang, B.Wei, Zagreb indices and multiplicative Zagreb indices of Eulerian graphs, Bull. Malays. Math. Sci. Soc. DOI: 10.1007/s40840-017-0463-2, to appear.
• [16] Z. Liu, Q. Ma, Y. Chen, New bounds on Zagreb indices, J. Math. Inequal. 11 (2017) 167- 179.
• [17] E. Munarini, C. P. Cippo, A. Scagliola, N. Z. Salvi, Double graphs, Discrete Math., 308 (2008), 242-254.
• [18] P. S. Ranjini, V. Lokesha, I. N. Cangul, On the Zagreb indices of the line graphs of the subdivision graphs, Appl. Math. Comput., 218 (2011), 699-702.
• [19] D. Sarala, H. Deng, S. K. Ayyaswamy, S. Balachandran, The Zagreb indices of graphs based on four new operations related to the lexicographic product, Appl. Math. Comp. 309 (2017) 156-169.
• [20] T. A. Selenge, B. Horoldagva, K. C. Das, Direct comparison of the variable Zagreb indices of cyclic graphs, MATCH Commun. Math. Comput. Chem. 78 (2017) 351-360.
• [21] R. Todeschini, V. Consonni, New local vertex invariant and molecular descriptors based on functions of the vertex degrees, MATCH, 64 (2010), 359-372.
• [22] M. Togan, A. Yurttas, I. N. Cangul, All versions of Zagreb indices and coindices of subdivision graphs of certain graph types, Advanced Studies in Contemporary Mathematics, 26 (1) (2016), 227-236.
• [23] M. Togan, A. Yurttas, I. N. Cangul, All versions of Zagreb indices and coindices of r-subdivision graphs of certain graph types (preprint).
• [24] M. Togan, A. Yurttas, I. N. Cangul, r-subdivision graphs of double graphs and their multiplicative Zagreb indices (preprint).
• [25] M. Togan, A. Yurttas, I. N. Cangul, Some formulae and inequalities on several Zagreb indices of r-subdivision graphs, Enlightments of Pure and Applied Mathematics (EPAM), 1 (1) (2015), 29-45.
• [26] M. Togan, A. Yurttas, I. N. Cangul, A. S. Cevik, Zagreb Indices and Multiplicative Zagreb Indices of Double Graphs of Subdivision Graphs (preprint).
• [27] D. Vukievi, J. Sedlar, D. Stevanovi, Comparing Zagreb indices for almost all graphs, MATCH Commun. Math. Comput. Chem. 78 (2017) 323-336.
• [28] A. Yurttas, M. Togan, I. N. Cangul, Zagreb indices and multiplicative Zagreb indices of subdivision graphs of double graphs, Advanced Studies in Contemporary Mathematics, 26 (3) (2016), 407-416.
• [29] A. Yurttas, M. Togan, A. S. Cevik, I. N. Cangul, Relations between the first and second Zagreb indices of subdivision graphs (preprint).