Abstract
The dendrimers are highly branched organic macromolecules having repeated iterations of branched units that surrounds the central core. Dendrimers are used in a variety of fields including chemistry, nanotechnology and biology. For positive integers \(n\)
and \(k\), the symmetric 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 \(k\) and all pendent vertices have equal distance \(n\) from the root.
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 Vargör and Dündar in 2011, of \(T_{n, k}\).