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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 ${K}_{k}\vee \left({K}_{1}\cup {K}_{n-k-1}\right)$ 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 ${K}_{\beta }\vee \overline{{K}_{n-\beta }}$.

##### MSC:
 05C40 Connectivity 05C05 Trees 05C90 Applications of graph theory 05C35 Extremal problems (graph theory)
##### References:
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