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

Document Type : Research Paper

Authors

1 Department of Mathematics and Computer Science‎, ‎Sirjan University of Technology‎, ‎Sirjan‎, ‎Iran

2 Department of Mathematics‎, ‎Kazerun Branch‎, ‎Islamic Azad University‎, ‎P‎. ‎O‎. ‎Box‎: ‎73135-168‎, Kazerun,‎ ‎Iran

Abstract

‎The Sombor index is a newly introduced vertex-degree-based graph invariant with the ability to predict the enthalpy‎
‎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
$$SO_{red}(\Gamma)=\sum_{\mathcal{FG}\inE(\Gamma)}\sqrt{(d_{\Gamma}(\mathcal{F})-1)^2+(d_{\Gamma}(\mathcal{G})-1)^2},$$
‎ and
$$SO^{\ddagger}(\Gamma)=\sum_{\mathcal{FG}\inE({\Gamma})}\sqrt{(d_{\Gamma}(\mathcal{F})+1)^2+(d_{\Gamma}(\mathcal{G})+1)^2},$$‎
‎ 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.

Keywords

Main Subjects

References

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Iranian Journal of Mathematical Chemistry
Volume 15, Issue 4
December 2024
Pages 227-237

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