on the multiplicative zagreb indices of bucket recursive‎ ‎trees

‎ bucket recursive trees are an interesting and natural generalization of ordinary recursive trees and have a connection to mathematical chemistry. in this paper, we give the lower and upper bounds for the moment generating function and moments of the multiplicative zagreb indices in a randomly chosen bucket recursive tree of size n with maximal bucket size b ≥ 1. also, we consider the ratio of the multiplicative zagreb indices for different values of n and b . all our results reduce to the ordinary recursive trees for b = 1 .