$l_1$-embeddability of pentagon-based open-ended nanotubes and its applications

A graph is $l_1$-embeddable if it admits a binary addressing such that the Hamming distance between the binary addresses is, up to scale, the distance in the graph between the corresponding vertices. The zigzag pentagon-based nanotubes and armchair pentagon-based nanotubes are two new materials with the tube shapes. In order to easily calculate their chemical indices based on distances, we showed that only five types $\mathcal{I}(p,1)$, $\mathcal{I}(1,q,\theta )$ with $q\ge 5$, $I(1,3,\theta )$, $I(2,3,\zeta )$ and $I(2,5,\zeta )$ zigzag pentagon-based nanotubes were $l_1$-embeddable. Finally, we gave the precise value of the Wiener index and the hyper-Wiener index of these five zigzag pentagon-based nanotubes.