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On the Zagreb index of random recursive trees. (English) Zbl 1234.05053

Summary: We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments of \(Z_{n}\), the Zagreb index of a random recursive tree of size \(n\), are obtained. We also show that the random process \({Z_{n} - E[Z_{n}], n \geq 1}\) is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by an application of the martingale central limit theorem. Finally, two other topological indices are also discussed in passing.

MSC:

05C05 Trees
05C80 Random graphs (graph-theoretic aspects)
05C10 Planar graphs; geometric and topological aspects of graph theory
60C05 Combinatorial probability
60F05 Central limit and other weak theorems
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