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 \Gamma as‎SO_{red}(\Gamma)=\sum_{\mathcal{FG}\inE (\Gamma)}\sqrt{(d_{\Gamma}(\mathcal{F})-1)^2+(d_{\Gamma}(\mathcal{G})-1)^2}, andSO^{\ddagger}(\Gamma)=\sum_{\mathcal{FG}\inE({\Gamma})}\sqrt{(d_{\Gamma}(\mathcal{F})+1)^2+(d_{\Gamma}(\mathcal{G})+1)^2},‎‎ in which d_{\Gamma}(\mathcal{F}) is the degree of the vertex \mathcal{F} in \ Gamma‎.‎ 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.