Further results on atom-bond connectivity index of trees. (English) Zbl 1216.05161

Summary: The atom-bond connectivity (ABC) index of a graph \(G\) is defined as \[ \text{ABC}(G)= \sum_{uv\in E(G)} \sqrt{{d_u+ d_v- 2\over d_u d_v}}, \] where \(E(G)\) is the edge set and \(d_u\) is the degree of vertex \(u\) of \(G\). We give the best upper bound for the ABC index of trees with a perfect matching, and characterize the unique extremal tree, which is a molecular tree. We also give upper bounds for the ABC index of trees with fixed number of vertices and maximum degree, and of molecular trees with fixed numbers of vertices and pendent vertices, and characterize the extremal trees.


05C90 Applications of graph theory
05C05 Trees
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