Wiener Indices of Spiro and Polyphenyl Hexagonal Chains

TL;DR

This paper derives formulas for Wiener indices of spiro and polyphenyl hexagonal chains, relates their indices, and identifies extremal structures, revealing that the average Wiener index equals that of the meta-chain.

Contribution

It provides explicit formulas and relations for Wiener indices of specific hexagonal chains, and characterizes extremal graphs within these classes.

Findings

Explicit formulas for Wiener indices of spiro and polyphenyl chains.

Relation between Wiener indices of spiro and polyphenyl chains.

Average Wiener index equals that of the meta-chain.

Abstract

The Wiener index W(G) of a connected graph $G$ is the sum of distances between all pairs of vertices in G$. In this paper, we first give the recurrences or explicit formulae for computing the Wiener indices of spiro and polyphenyl hexagonal chains, which are graphs of a class of unbranched multispiro molecules and polycyclic aromatic hydrocarbons, then we establish a relation between the Wiener indices of a spiro hexagonal chain and its corresponding polyphenyl hexagonal chain, and determine the extremal values and characterize the extremal graphs with respect to the Wiener index among all spiro and polyphenyl hexagonal chains with n hexagons, respectively. An interesting result shows that the average value of the Wiener indices with respect to the set of all such hexagonal chains is exactly the average value of the Wiener indices of three special hexagonal chains, and is just the Wiener index of the meta-chain.