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.