Counting the numbers of paths of all lengths in dendrimers and its applications

Abstract

For positive integers n and k, the dendrimer T n,k is defined as the rooted tree of radius n whose all vertices at distance less than n from the root have degree k. The dendrimers are higly branched organic macromolecules having repeated iterations of branched units that surroundes the central core. Dendrimers are used in a variety of fields including chemistry, nanotechnology, biology. In this paper, for any positive integer ℓ, we count the number of paths of length ℓ of T n,k . As a consequence of our main results, we obtain the average distance of T n,k which we can establish an alternate proof for the Wiener index of T n,k . Further, we generalize the concept of medium domination, introduced by Vargör and Dündar in 2011, of T n,k .