  Veit Elser proposed a random graph model for percolation in which physical
dimension appears as a parameter. Studying this model combinatorially leads
naturally to the consideration of numerical graph invariants which we call
\emph{Elser numbers} $\mathsf{els}_k(G)$, where $G$ is a connected graph and
$k$ a nonnegative integer. Elser had proven that $\mathsf{els}_1(G)=0$ for all
$G$. By interpreting the Elser numbers as Euler characteristics of appropriate
simplicial complexes called \emph{nucleus complexes}, we prove that for all
graphs $G$, they are nonpositive when $k=0$ and nonnegative for $k\geq2$. The
last result confirms a conjecture of Elser. Furthermore, we give necessary and
sufficient conditions, in terms of the 2-connected structure of~$G$, for the
nonvanishing of the Elser numbers.
