Dorpalen-Barry, GalenHettle, CyrusLivingston, David C.Martin, Jeremy L.Nasr, George D.Vega, JulianneWhitlatch, Hays2021-02-152021-02-152020-12-18Galen Dorpalen-Barry, Cyrus Hettle, David C. Livingston, Jeremy L. Martin, George D. Nasr, Julianne Vega, Hays Whitlatch, "A positivity phenomenon in Elser's Gaussian-cluster percolation model", Journal of Combinatorial Theory, Series A, Volume 179, 2021, 105364, ISSN 0097-3165, https://doi.org/10.1016/j.jcta.2020.105364.https://hdl.handle.net/1808/31427Veit 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.© 2020 Elsevier Inc. All rights reserved. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.http://creativecommons.org/licenses/by-nc-nd/4.0/GraphSimplicial complexEuler characteristicNucleusPercolationBlock-cutpoint treeArticle10.1016/j.jcta.2020.105364embargoedAccess