RW index of the random polyphenylene chains

For a connected graph G , the RW index is defined as the sum of the expected walks lengths between all pairs of vertices in the graph G . This article is devoted to establish the explicit analytical expressions for the expected value of the RW index of a random polyphenylene chain. Furthermore, the average value and the sample variance of the RW index with respect to the set of all polyphenylene chains with n hexagons are obtained. The correlation between RW index and Wiener index of the random polyphenylene chain is also characterized. Finally, we prove the RW index of the random polyphenylene chain is asymptotic to normal distributions.