We also identify and describe the graphs that achieve extremal values.
📌 3 Key Takeaways
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Objective & Hypothesis:The atom-bond sum-connectivity index of a graph $G$ is defined as \begin{equation*} ABSC(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{d(u)+d(v)-2}{d(u)+d(v)}}=\sum\limits_{uv\in E(G)}\sqrt{1-\frac{2}{d(u)+d(v)}} \end{equation*} where $d(u)$ and $d(v)$ represent the degrees of vertices $u$ and $v$ respectively.
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Key Findings & Data:In this paper, we will determine the sharp upper and lower bounds for the atom-bond sum-connectivity index of a series of graph operations, including the addition $\left(G+H\right)$, the Cartesian product$\left(G\times H\right)$, the composition $\left(G[H]\right)$, and the corona product $\left(GoH\right)$ of graphs, where $G$ and $H$ representing graphs.
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Conclusion & Impact:We also identify and describe the graphs that achieve extremal values.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:ABSC index · graph operations · sharp inequalities
Abstract
The atom-bond sum-connectivity index of a graph $G$ is defined as \begin{equation*} ABSC(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{d(u)+d(v)-2}{d(u)+d(v)}}=\sum\limits_{uv\in E(G)}\sqrt{1-\frac{2}{d(u)+d(v)}} \end{equation*} where $d(u)$ and $d(v)$ represent the degrees of vertices $u$ and $v$ respectively. In this paper, we will determine the sharp upper and lower bounds for the atom-bond sum-connectivity index of a series of graph operations, including the addition $\left(G+H\right)$, the Cartesian product$\left(G\times H\right)$, the composition $\left(G[H]\right)$, and the corona product $\left(GoH\right)$ of graphs, where $G$ and $H$ representing graphs. We also identify and describe the graphs that achieve extremal values.
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