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

Document Type : Original paper


Sirjan University of Technology, Sirjan 78137, Iran


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. ‎‎


Main Subjects

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