On the First Variable Zagreb Index

Document Type : Research Paper


1 Department of Statistics, Islamic Azad University

2 Imam Khomeini international university


‎The first variable Zagreb index of graph $G$ is defined as‎
‎M_{1,\lambda}(G)=\sum_{v\in V(G)}d(v)^{2\lambda}‎,
‎where $\lambda$ is a real number and $d(v)$ is the degree of‎
‎vertex $v$‎.
‎In this paper‎, ‎some upper and lower bounds for the distribution function and expected value of this index in random increasing trees (recursive trees‎,
‎plane-oriented recursive trees and binary increasing trees) are‎


Main Subjects

1. V. Andova and M. Petrusevski, Variable Zagreb indices and Karamata’s inequality,
MATCH Commun. Math. Comput. Chem. 65 (2011) 685–690.
2. M. Drmota, Random Trees, An Interplay Between Combinatorics and Probability,
Springer, Wien-New York, 2009.
3. R. Kazemi, Probabilistic analysis of the first Zagreb index, Trans. Comb. 2 (2)
(2013) 35–40.
4. R. Kazemi, The eccentric connectivity index of bucket recursive trees, Iranian J.
Math. Chem. 5 (2) (2014) 77–83.
5. R. Kazemi, The second Zagreb index of molecular graphs with tree structure,
MATCH Commun. Math. Comput. Chem. 72 (2014) 753–760.
6. R. Kazemi and L. K. Meimondari, Degree distance and Gutman index of increasing
trees, Trans. Comb. 5 (2) (2016) 23–31.
7. M. Kuba and A. Panholzer, On the degree distribution of the nodes in increasing
trees, J. Combin. Theory Ser. A 114 (4) (2007) 597–618.
8. J. Szymanski, On the maximum degree and height of a random recursive tree,
Wiley, New York, 1990.