The first Zagreb and forgotten topological indices of d-ary trees

Year 2017, Volume: 46 Issue: 4, 603 - 611, 01.08.2017

Ramin Kazemi Ali Behtoei

Abstract

The first Zagreb index of a graph is equal to the sum of the square of the vertex degrees of the graph and the forgotten topological index (Findex) of a graph is defined as the sum of cubes of the vertex degrees of the graph. These parameters have applications in chemistry and drug structures. For any fixed integer $d\geq2$, the d-ary tree is a rooted tree in which each node has no more than d children. We determine the mean value of these indices in d-ary trees. Also, we obtain some useful relations and equalities related to these indices and other graphical parameters. 

Keywords

First Zagreb index , Forgotten topological index , d-ary tree , Mean

References

• Balaban, A.T. and Motoc, I. and Bondchev, D. and Mekenyan, O. Topological indices for structure-activity correlations, Topics Curr. Chem. 114, 21-55, 1983.
• Che, Z. and Chen, Z. Lower and upper bounds of the forgotten topological index, MATCH Commun. Math. Comput. Chem. 76 (3), 635-648, 2016.
• Das, K.C. and Gutman, I. Bounds for the energy of graphs, Hacet. J. Math. Stat. 45 (3), 695-703, 2016.
• Dosli¢, T. and Réti, T. and Vukicevi¢, D. On the vertex degree indices of connected graphs, Chem. Phys. Lett. 512, 283-286, 2011.
• Feng, Q. and Hu, Z. On the Zagreb index of random recursive trees, J. Appl. Prob. 48 1189-1196, 2011.
• Feng, Q. and Hu, Z. Asymptotic normality of the Zagreb index of random b-ary recursive trees, Far Eastern Math. J. 15(1) 91-101, 2015.
• Furtula, B. and Gutman, I. A forgotten topological index, J. Math. Chem. 53 (4), 1184-1190, 2015.
• Gao, W. and Farahani, M.R. and Shi, L. Forgotten topological index of some drug structures, Acta Med. Medit., in press, 2016.
• Gutman, I. and Furtula, B. Estimating the second and third geometric-arithmetic indices, Hacet. J. Math. Stat. 40 (1), 69-76 , 2011.
• Gutman, I. and Trinajsti¢, N. Graph theory and molecular orbitals. Total -electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17, 535-538, 1972.
• Horoldagva, B. and Das, K.C. On comparing Zagreb indices of graphs, Hacet. J. Math. Stat. 41 (2), 223-230, 2012.
• Kazemi, R. The second Zagreb index of molecular graphs with tree structure, MATCH Commun. Math. Comput. Chem. 72, 753760, 2014.
• Sun, Y.C. and Lin, Z. and Peng, W.X. and Yuan, T.Q. and Xu. F. and Wu, Y.Q. and Yang, J. and Wang, Y.S. and Sun, R.C. Chemical changes of raw materials and manufactured binderless boards during hot pressing: lignin isolation and characterization, Bioresources 9 (1), 1055-1071, 2014.
• Yang, L. and Ai, X. and Zhang, L. The Zagreb coindices of a type of composite graph, Hacet. J. Math. Stat. 45 (4), 1135-1142, 2016.