a ‎note‎ ‎on‎ ‎the‎ ‎r‎e-defined third zagreb index of trees

For a graph $gamma$‎, ‎the re-defined third zagreb index is defined as $$rezg_3(gamma)=sum_{abin e(gamma)}deg_gamma(a) ‎deg_gamma(b)big(‎deg_gamma(a)+‎deg_gamma(b)big)‎‎,$$‎‎where $deg_gamma(a)$ is the degree of‎ ‎vertex $a$‎. ‎we prove for any tree $t$ with $n$ vertices and maximum degree $delta$‎, ‎‎$rezg_3(t)geq‎16n+delta^3+2delta^2-13delta-26$ ‎when ‎‎$‎delta< n-1‎$ ‎and‎ $rezg_3(t)=‎ndelta^2+ndelta-delta^2-delta$ ‎when ‎‎$‎delta=n-1‎$. ‎also we determine the corresponding extremal trees‎. ‎‎