Multiplicative Sombor index of trees

For a graph, the multiplicative Sombor index is defined as ₒ₎ () =₀₁ ₄ () d²_ (a) +d²_ (b), where d_ (a) is the degree of vertex a. Liu Liu, H. (2022). Discrete Mathematics Letters, 9, 80–85 showed that, when T is a tree of order n, ₒ₎ (T) ₒ₎ (Pₙ) =5 (8) ^n-3. We improved this result and show that, if T is a tree of order n with maximum degree D, then ₒ₎ (T) \array{ll (5 (D²+4) ) ^{D2}8^n-2{{D-1}2} D-12, \\2mm (D²+1) ^2{{D+1-n}2} (5 (D²+4) ) ^n-{{D-1}2} D>n-12. array. Also, we show that equality holds if and only if T is a spider whose all legs have length less than three or all legs have length more than one.