A ‎note‎ ‎on‎ ‎the‎ ‎r‎e-defined third Zagreb index of trees

Author(s):

Nasrin Dehgardi *

Message:
Article Type:
Research/Original Article (دارای رتبه معتبر)
Abstract:
For a graph $\Gamma$‎, ‎the re-defined third Zagreb index is defined as $$ReZG_3(\Gamma)=\sum_{ab\in E(\Gamma)}\deg_\Gamma(a) ‎\deg_\Gamma(b)\Big(‎\deg_\Gamma(a)+‎\deg_\Gamma(b)\Big)‎‎,$$‎‎where $\deg_\Gamma(a)$ is the degree of‎ ‎vertex $a$‎. ‎We prove for any tree $T$ with $n$ vertices and maximum degree $\Delta$‎, ‎‎$ReZG_3(T)\geq‎16n+\Delta^3+2\Delta^2-13\Delta-26$ ‎when ‎‎$‎\Delta< n-1‎$ ‎and‎ $ReZG_3(T)=‎n\Delta^2+n\Delta-\Delta^2-\Delta$ ‎when ‎‎$‎\Delta=n-1‎$. ‎Also we determine the corresponding extremal trees‎. ‎‎
Keywords:

Zagreb ‎ Indices , ‎ Re-Defined Third Zagreb Index , ‎ ‎ Trees

Language:
English
Published:
Communications in Combinatorics and Optimization, Volume:10 Issue: 3, Summer 2025
Pages:
539 to 545
https://www.magiran.com/p2814617