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Title:
The General Sombor Index of Extremal Trees with a Given Maximum Degree
Authors:
Sultan Ahmad, Rashid Farooq, Kinkar Chandra Das
Volume
94
Issue
3
Year
2025
Pages
825-853
Abstract

The general Sombor index of a graph \(G\), denoted by \({\cal {SO}}_{\alpha}(G)\), is recently defined as: \begin{align*} {\cal {SO}}_{\alpha}(G)=\sum_{ \substack{v_iv_j\in E(G)}}\,\Big(d_G(v_i)^2+d_G(v_j)^2\Big)^{\alpha}, \end{align*} where \(d_G(v_i)\) represents the degree of vertex \(v_i\), and \(\alpha\) is an arbitrary real number. This study focuses on identifying extremal trees for the general Sombor index within the class of \(n\)-vertex trees with maximum degree \(\Delta\). We analyze the general Sombor index across various intervals of \(\alpha\). Specifically, for \(\alpha>1\) and \(\alpha\in [-1,0)\), we determine the trees that maximize the general Sombor index. Moreover, for \(\alpha<0\) and \(\alpha>0\), we identify the trees that minimize the general Sombor index \({\cal {SO}}_{\alpha}\). Finally, the characterization of extremal trees for \({\mathcal {SO}}_{\alpha}\) in the remaining intervals of \(\alpha\) remains an open problem and presents a promising direction for future research.