Some Topological Invariants of Generalized M\"obius Ladder

The Hosoya polynomial of a graph $G$ was introduced by H. Hosoya in 1988 as a counting polynomial, which actually counts the number of distances of paths of different lengths in $G$. The most interesting application of the Hosoya polynomial is that almost all distance-based graph invariants, which are used to predict physical, chemical and pharmacological properties of organic molecules, can be recovered from it. In this article we give the general closed form of the Hosoya polynomial of the generalized M\"obius ladder $M(m,n)$ for arbitrary $m$ and for $n=3$. Moreover, we recover Wiener, hyper Wiener, Tratch-Stankevitch-Zefirov, and Harary indices from it.