Net-degree variance and Sombor index of signed graphs

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Document Type : Original paper

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South Asian University, New Delhi-110068, India

Abstract

A \textit{signed graph} Σ is an ordered pair (Σu,σ), where Σu=(V,E) is the \textit{underlying graph} and σ is sign mapping called \textit{signature}, which assigns each edge in E a sign from the set {+,−}. The study of vertex-degree-based topological index: known as \textit{Sombor index} was initiated by I. Gutman in 2021 for any graph G. He defined it as SO(G)=∑eij∈E(G)dG(vi)2+dG(vj)2. In this work, the concept of the Sombor index is extended to connected signed graphs. The Sombor index is derived mathematically for signed paths and signed cycles, and is supported by computational algorithms. Furthermore, it is proved that the Sombor index of a connected signed graph Σ is maximized if and only if the \textit{net-degree variance} of Σ is also maximized. As an application, this study provides a solution to the net-degree variance maximization problem for certain types of signed graphs.\

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 20 May 2025

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