  The Shor-Laflamme distribution (SLD) of a quantum state is a collection of
local unitary invariants that quantify $k$-body correlations. We show that the
SLD of graph states can be derived by solving a graph-theoretical problem. In
this way, the mean and variance of the SLD are obtained as simple functions of
efficiently computable graph properties. Furthermore, this formulation enables
us to derive closed expressions of SLDs for some graph state families. For
cluster states, we observe that the SLD is very similar to a binomial
distribution, and we argue that this property is typical for graph states in
general. Finally, we derive an SLD-based entanglement criterion from the purity
criterion and apply it to derive meaningful noise thresholds for entanglement.
Our new entanglement criterion is easy to use and also applies to the case of
higher-dimensional qudits. In the bigger picture, our results foster the
understanding both of quantum error-correcting codes, where a closely related
notion of Shor-Laflamme distributions plays an important role, and of the
geometry of quantum states, where Shor-Laflamme distributions are known as
sector length distributions.
