Lower Bounds on the Entire Sombor Index

Let G = (V, E) be a graph. The entire Sombor index of graph G, SO epsilon (G) is defined as the sum of the terms root d(G)(2)(a) + d(G)(2)(b), where a is either adjacent to or incident with b and a, b is an element of V boolean OR E. It is known that if T is a tree of order n, then SO epsilon (T) >= 6 root 5 + 8(n - 3)root 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.