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

## TL;DR

This paper investigates Elser's Gaussian-cluster percolation model, introducing Elser numbers as graph invariants, proving their sign properties, and confirming a conjecture about their behavior related to graph connectivity.

## Contribution

The paper connects Elser numbers to Euler characteristics of nucleus complexes and proves their sign properties, confirming Elser's conjecture and characterizing when they are nonzero.

## Key findings

- Elser numbers are nonpositive for k=0 and nonnegative for k≥2.
- Confirmed Elser's conjecture on the sign of Elser numbers.
- Provided conditions based on 2-connected structure for nonvanishing Elser numbers.

## 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 \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.