The fractional chromatic number of double cones over graphs

Assume $n, m$ are positive integers and $G$ is a graph. Let $P_{n,m}$ be the graph obtained from the path with vertices $\{-m, -(m-1), \ldots, 0, \ldots, n\}$ by adding a loop at vertex $0$. The double cone $\Delta_{n,m}(G)$ over a graph $G$ is obtained from the direct product $G \times P_{n,m}$ by identifying $V(G) \times \{n\}$ into a single vertex $(\star, n)$, identifying $V(G) \times \{-m\}$ into a single vertex $(\star, -m)$, and adding an edge connecting $(\star, -m)$ and $(\star, n)$. This paper determines the fractional chromatic number of $\Delta_{n,m}(G)$. In particular, if $n < m$ or $n=m$ is even, then $\chi_f(\Delta_{n,m}(G)) = \chi_f(\Delta_n(G))$, where $\Delta_n(G)$ is the $n$th cone over $G$. If $n=m$ is odd, then $\chi_f(\Delta_{n,m}(G)) > \chi_f(\Delta_n(G))$. The chromatic number of $\Delta_{n,m}(G)$ is also discussed.