  We describe a construction that maps any connected graph G on three or more
vertices into a larger graph, H(G), whose independence number is strictly
smaller than its Lov\'asz number which is equal to its fractional packing
number. The vertices of H(G) represent all possible events consistent with the
stabilizer group of the graph state associated with G, and exclusive events are
adjacent. Mathematically, the graph H(G) corresponds to the orbit of G under
local complementation. Physically, the construction translates into
graph-theoretic terms the connection between a graph state and a Bell
inequality maximally violated by quantum mechanics. In the context of
zero-error information theory, the construction suggests a protocol achieving
the maximum rate of entanglement-assisted capacity, a quantum mechanical
analogue of the Shannon capacity, for each H(G). The violation of the Bell
inequality is expressed by the one-shot version of this capacity being strictly
larger than the independence number. Finally, given the correspondence between
graphs and exclusivity structures, we are able to compute the independence
number for certain infinite families of graphs with the use of quantum
non-locality, therefore highlighting an application of quantum theory in the
proof of a purely combinatorial statement.
