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

Abstract

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 Elser numbers els_k(G), where G is a connected graph and k a nonnegative integer. Elser had proven that els_1(G) = 0 for all G. By interpreting the Elser numbers as reduced Euler characteristics of appropriate simplicial complexes called nucleus complexes, we prove that for all graphs G, they are nonpositive when k = 0 and nonnegative for k ≥ 2. 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.