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 \(T_{n, k}\).