lower bounds on the entire sombor index

Let g = (v;e) be a graph. the entire sombor index opf graph g, so(g) is defined as the sum of the terms d2 g(a) + d2 g(b), where a is either adjacent to or incident with b and a; b 2 v [ e. it is known that if t is a tree of order n, then so(t)  6 p 5 + 8(n 􀀀 3) p 2. we improve this result and establish best lower bounds on the entire sombor index with given vertices number and maximum degree. also, we determine the extremal trees achieve these bounds.