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

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 $\ell$, we count the number of paths of length $\ell$ 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 Varg\"{o}r and D\"{u}ndar in 2011, of $T_{n, k}$.