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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. {\it B. Furtula}, {\it A. Graovac} and {\it 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 $K_\beta \vee \overline {K_{n-\beta}}$.

05C90Applications of graph theory
05C35Extremal problems (graph theory)
Full Text: DOI
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