in this paper we study the zagreb index in bucket recursive trees containing buckets with variable capacities. this model was introduced by kazemi in 2012. weobtain the mean and variance of the zagreb index andintroduce a martingale based on this quantity.

Imagine a graph which is progressively destroyed by cutting its edges one after the other in a uniform random order. The so-called cut-tree records key steps of this destruction process. It can be viewed as a random metric space equipped with a natural probability mass. In this work, we show that the cut-tree of a random recursive tree of size n, rescaled by the factor n−1 lnn, converges in pro...

In this article, we investigate the Zagreb index, a kind of graph-based topological several random networks, including class networks extended from recursive trees, plain-oriented and caterpillars growing in preferential attachment manner. We calculate mean variance index for each class. addition, prove that asymptotic distribution first is normal, second skewed to right.