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The reduced Sombor index and the exponential reduced Sombor index of a molecular tree
  • Fangxia Wang,
  • Baoyindureng Wu
Fangxia Wang
Xinjiang University
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Baoyindureng Wu
Xinjiang University

Corresponding Author:[email protected]

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Abstract

The reduced Sombor index $SO_{red}(G)$ and the exponential reduced Sombor index of a graph $G$ $e^{SO_{red}}(G)$ are defined respectively as $$SO_{red}(G)=\sum_{uv\in E(G)}\sqrt{(d(u)-1)^2+(d(v)-1)^2},$$ $$e^{SO_{red}}(G)=\sum_{uv\in E(G)}e^{\sqrt{(d(u)-1)^2+(d(v)-1)^2}},$$ where $d(u)$ denotes the degree of the vertex $u$ in $G$. In this paper, we obtain the maximum value of the reduced Sombor index among all molecular trees of order $n$ with perfect matching and characterize the corresponding extremal trees. This sloves a problem of the reduced Sombor index posed by Deng, Tang and Wu (Molecular trees with extremal values of Sombor indices, International Journal of Quantum Chemistry, 121 (2021) e26622). We also show that the maximum molecular trees of exponential reduced Sombor index and reduced Sombor index are the same, which was conjectured by Liu, You, Tang and Liu (On the reduced Sombor index and its applications, MATCH Commun. Math. Comput. Chem. 86 (2021) 729-753).