Comparison of the Wiener and Kirchhoff Indices of Random Pentachains

Abstract

Let $G$ be a connected (molecule) graph. The Wiener index $W\left(G\right)$ and Kirchhoff index $Kf\left(G\right)$ of $G$ are defined as the sum of distances and the resistance distances between all unordered pairs of vertices in $G$ , respectively. In this paper, explicit formulae for the expected values of the Wiener and Kirchhoff indices of random pentachains are derived by the difference equation and recursive method. Based on these formulae, we then make comparisons between the expected values of the Wiener index and the Kirchhoff index in random pentachains and present the average values of the Wiener and Kirchhoff indices with respect to the set of all random pentachains with $n$ pentagons.