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.