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.