A positivity phenomenon in Elser's Gaussian-cluster percolation model

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.