Lower and Upper Bounds between Energy‎, ‎Laplacian Energy‎, ‎and Sombor Index of Some Graphs

Document Type : Research Paper

Author

Department of Mathematics‎, ‎Tafresh University‎, ‎Tafresh 39518-79611‎, ‎Iran

10.22052/ijmc.2024.254674.1850

Abstract

‎Ivan Gutman has introduced two essential indices; the energy of a graph G‎, ‎and the Sombor index of that‎. ‎$\varepsilon(G)$‎, ‎which stands for the first index‎, ‎is the sum of the absolute values of all eigenvalues related to the adjacency matrix of the graph $G$‎. ‎The second‎, ‎defined as $SO(G)=\sum _{uv \in E(G)}\sqrt{d_u^2+d_v^2}$‎, ‎where $d_u$ and $d_v$ are the degrees of vertices $u$ and $v$ in $G$‎, ‎respectively‎. ‎It was proved that if $G$ is a graph of order at least 3‎, ‎then $\varepsilon(G)\leq So(G)$ and if $G$ is a connected graph of order $n$ that is not $P_n$ for $n\leq 8$‎, ‎then $\varepsilon(G)\leq \frac{So(G)}{2}$‎.
‎In this paper‎, ‎we have strengthened these results and will obtain several lower and upper bounds between the energy of a graph‎, ‎Laplacian energy‎, ‎and the Sombor index‎.

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References

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Iranian Journal of Mathematical Chemistry
Volume 16, Issue 1
In progress
March 2025
Pages 33-38

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