On the Reduced and Increased Sombor Indices of Trees‎ ‎with‎ ‎Given‎ ‎Order and Maximum‎ ‎Degree

‎The Sombor index is a newly introduced vertex-degree-based graph invariant with the ability to predict the enthalpy‎‎of vaporization and entropy of octane isomers‎. ‎Recently‎, ‎two new variants of the Sombor index namely the reduced and increased Sombor indices were put forward‎. ‎The reduced and increased Sombor indices are respectively defined for graph Γ as SOred(Γ) = ∑ FG∈E(Γ) √ (dΓ(F) − 1)² + (dΓ(G) − 1)², and SO‡ (Γ) = ∑ FG∈E(Γ) √ (dΓ(F) + 1)² + (dΓ(G) + 1)² , in which dΓ(F) is the degree of the vertex F in Γ. Our purpose is to establish sharp lower bounds on the reduced and increased Sombor indices of trees in terms of their order and maximum vertex degree‎. Moreover‎, ‎the extremal trees that attain the bounds are characterized‎.