We also identify and describe the graphs that achieve extremal values.
📌 3 Maddede Bu Makale
🎯
Amaç & Hipotez: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.
📊
Temel Bulgular & Veri: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.
💡
Sonuç & Katkı:We also identify and describe the graphs that achieve extremal values.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler: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.
Hussain, Z. (2026). Sharp inequalities for the atom-bond sum-connectivity index of graph operations. Hacettepe Journal of Mathematics and Statistics, Volume: 55 Issue: 1. https://doi.org/10.15672/hujms.1456544
A. Ali, B. Furtula, I. Redžepovic and I. Gutman, Atom-bond sum-connectivity index, J. Math. Chem. 60, 2081-2093, 2022.
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.
M.V. Diudea, I. Gutman and L. Jantschi, Molecular Topology, Huntington, NY, 2001.
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.
Y. Ge, Z. Lin and J. Wang, Atom-bond sum-connectivity index of line graphs, Discrete Math. Lett. 12, 196–200, 2023.
A. Graovac and T. Pisanski, On the Wiener index of a graph, J. Math. Chem. 8, 53–62, 1991.
S. Hossein-Zadeh, A. Hamzeh and A.R. Ashrafi, Wiener-type invariants of some graph operations, Filomat 23 (3), 103–113, 2009.
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.
W. Imrich and S. Klavzar, Product graphs: structure and recognition, John Wiley & Sons, New York, USA, 2000.
S. Klavžar, A. Rajapakse and I. Gutman, The Szeged and the Wiener index of graphs, Appl. Math. Lett. 9, 45–49, 1996.
S. Klavžar, On the PI index: PI-partitions and Cartesian product graphs, MATCH Commun. Math. Comput. Chem. 57, 573–586, 2007.
M. Tavakoli, F. Rahbarnia, A.R. Ashrafi, Studying the corona product of graphs under some graph invariants, Trans. Comb. 3 (3), 43–49, 2014.
D.B. West, Introduction to Graph Theory, Prentice-Hall, Upper Saddle River, NJ, 1996.
Y.N. Yeh, I. Gutman, On the sum of all distances in composite graphs, Discrete Math. 135, 359–365, 1994.