University of Kashan Iranian Journal of Mathematical Chemistry 2228-6489 15 4 2024 12 01 On the Reduced and Increased Sombor Indices of Trees‎ ‎with‎ ‎Given‎ ‎Order and Maximum‎ ‎Degree 227 237 114549 10.22052/ijmc.2024.254548.1845 EN Nasrin Dehgardi Department of Mathematics and Computer Science‎, ‎Sirjan University of Technology‎, ‎Sirjan‎, ‎Iran Mahdieh Azari Department of Mathematics‎, ‎Kazerun Branch‎, ‎Islamic Azad University‎, ‎P‎. ‎O‎. ‎Box‎: ‎73135-168‎, Kazerun,‎ ‎Iran 0000-0002-0919-0598 Journal Article 2024 03 13 ‎The Sombor index is a newly introduced vertex-degree-based graph invariant with the ability to predict the enthalpy‎<br />‎of 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 $\Gamma$ as‎<br />$$SO_{red}(\Gamma)=\sum_{\mathcal{FG}\inE(\Gamma)}\sqrt{(d_{\Gamma}(\mathcal{F})-1)^2+(d_{\Gamma}(\mathcal{G})-1)^2},$$<br />‎ and<br />$$SO^{\ddagger}(\Gamma)=\sum_{\mathcal{FG}\inE({\Gamma})}\sqrt{(d_{\Gamma}(\mathcal{F})+1)^2+(d_{\Gamma}(\mathcal{G})+1)^2},$$‎<br />‎ in which $d_{\Gamma}(\mathcal{F})$ is the degree of the vertex $\mathcal{F}$ in $\Gamma$‎.<br />‎ 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. https://ijmc.kashanu.ac.ir/article_114549_2d296199b34af6c2e4ade4f9fd03a7eb.pdf