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

TL;DR

This paper develops formulas for counting paths of all lengths in dendrimers, enabling analysis of their average distance, Wiener index, and a generalized medium domination concept, with applications in chemistry and nanotechnology.

Contribution

It provides a comprehensive method to count paths of all lengths in dendrimers and extends the concept of medium domination to these structures.

Findings

Derived formulas for path counts of all lengths in dendrimers.

Established an alternative proof for the Wiener index of dendrimers.

Generalized the medium domination concept for dendrimers.

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 $\ell$, 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 $T_{n, k}$.