Schultz Polynomial and Modified Schultz Polynomial of a Random Benzenoid Chain

Abstract

Let \(G\) be a graph with vertex set \(V(G)\), \(d_G(u,v)\) and \(\delta_G(v)\) denoteas the topological distance between vertices \(u\) and \(v\) in \(G\), and \(d_G(v)\) as the degree of vertex \(v\) in \(G\),respectively. The Schultz polynomial of \(G\) is defined as \(H^+(G) = \sum\limits_{u,v \subseteq V(G)} (\delta _G(u)+\delta _G(v))x^{d_G(u,v)}\), and the modified Schultz polynomial of \(G\) is defined as \(H^*(G) = \sum\limits_{u,v \subseteq V(G)}(\delta _G(u)+\delta _G(v)) x^{d_G(u,v)}\). In this paper, we obtain explicit analytical expressions for the expected values of the Schultz polynomial and modified Schultz polynomial of a random benzenoid chain with $n$ hexagons. Furthermore, we derive expected values of some related topological indices.