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

Galen Dorpalen-Barry, University of Minnesota Twin Cities
Cyrus Hettle, Georgia Institute of Technology
David C. Livingston, University of Wyoming
Jeremy L. Martin, University of Kansas
George D. Nasr, University of Nebraska–Lincoln
Julianne Vega, Kennesaw State University
Hays Whitlatch, Gonzaga University


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 elsk(G), where G is a connected graph and k a nonnegative integer. Elser had proven that els1(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.