Some upper bounds for the atom-bond connectivity index of graphs. (English) Zbl 1246.05091

Summary: The recently introduced atom-bond connectivity (ABC) index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes. B. Furtula, A. Graovac and D. Vukičević [Discrete Appl. Math. 157, No. 13, 2828–2835 (2009; Zbl 1209.05252)] determined the extremal ABC values for chemical trees, and showed that the star tree \(S_{n}\) has the maximal ABC index among all trees.
In this work, we show that among all n-vertex graphs with vertex connectivity \(k\), the graph \(K_{k} \vee (K_{1}\cup K_{n - k - 1})\) is the unique graph with maximum ABC index. Furthermore, we determine the maximum ABC index of a connected graph with n vertices and matching number \(\beta \) and characterize the unique extremal graph as \(K_\beta \vee \overline {K_{n-\beta}}\).


05C40 Connectivity
05C05 Trees
05C90 Applications of graph theory
05C35 Extremal problems in graph theory


Zbl 1209.05252
Full Text: DOI


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