Sharp inequalities for the atom-bond sum-connectivity index of graph operations

Year 2026, Volume: 55 Issue: 1, 60 - 70, 23.02.2026

Zaryab Hussain

https://doi.org/10.15672/hujms.1456544

https://izlik.org/JA85JF48ZY

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.

Keywords

ABSC index , graph operations , sharp inequalities , extremal bounds

Ethical Statement

This work is done by me. No AI used for this manuscript.

References

• [1] A. Ali, B. Furtula, I. Redžepovic and I. Gutman, Atom-bond sum-connectivity index, J. Math. Chem. 60, 2081-2093, 2022.
• [2] A. Ali, I. Gutman and I. Redžepovic, Atom-bond sum-connectivity index of unicyclic graphs and some applications, Electron. J. Math. 5, 1–7, 2023.
• [3] M.V. Diudea, I. Gutman and L. Jantschi, Molecular Topology, Huntington, NY, 2001.
• [4] G.H. Fath-Tabar, B. Vaez-Zadeh, A.R. Ashrafi and A. Graovac, Some inequalities for the atom-bond connectivity index of graph operations, Discrete Appl. Math. 159 (13), 1323–1330, 2011.
• [5] Y. Ge, Z. Lin and J. Wang, Atom-bond sum-connectivity index of line graphs, Discrete Math. Lett. 12, 196–200, 2023.
• [6] A. Graovac and T. Pisanski, On the Wiener index of a graph, J. Math. Chem. 8, 53–62, 1991.
• [7] S. Hossein-Zadeh, A. Hamzeh and A.R. Ashrafi, Wiener-type invariants of some graph operations, Filomat 23 (3), 103–113, 2009.
• [8] Z. Hussain, H. Liu and H. Hua, Bounds for the atom-bond sum-connectivity index of graphs, MATCH Commun. Math. Comput.Chem. 94 (3), 805–824, 2025.
• [9] W. Imrich and S. Klavzar, Product graphs: structure and recognition, John Wiley & Sons, New York, USA, 2000.
• [10] S. Klavžar, A. Rajapakse and I. Gutman, The Szeged and the Wiener index of graphs, Appl. Math. Lett. 9, 45–49, 1996.
• [11] S. Klavžar, On the PI index: PI-partitions and Cartesian product graphs, MATCH Commun. Math. Comput. Chem. 57, 573–586, 2007.
• [12] M. Tavakoli, F. Rahbarnia, A.R. Ashrafi, Studying the corona product of graphs under some graph invariants, Trans. Comb. 3 (3), 43–49, 2014.
• [13] D.B. West, Introduction to Graph Theory, Prentice-Hall, Upper Saddle River, NJ, 1996.
• [14] Y.N. Yeh, I. Gutman, On the sum of all distances in composite graphs, Discrete Math. 135, 359–365, 1994.