  Given a weighted graph $G_\bx$, where $(x(v): v \in V)$ is a non-negative,
real-valued weight assigned to the vertices of G, let $B(G_\bx)$ be an upper
bound on the fractional chromatic number of the weighted graph $G_\bx$; so
$\chi_f(G_\bx) \le B(G_\bx)$. To investigate the worst-case performance of the
upper bound $B$, we study the graph invariant $$\beta(G) = \sup_{\bx \ne 0}
\frac{B(G_\bx)}{\chi_f(G_\bx)}.$$
  \noindent This invariant is examined for various upper bounds $B$ on the
fractional chromatic number. In some important cases, this graph invariant is
shown to be related to the size of the largest star subgraph in the graph. This
problem arises in the area of resource estimation in distributed systems and
wireless networks; the results presented here have implications on the design
and performance of decentralized communication networks.
