The Griffiths-Hurst-Sherman inequality for exponential random graphs: a counterexample to a conjecture and the case of chosen graphs with at most two edges

An exponential random graph is a Gibbs measure on the graphs with n labeled vertices. Its Hamiltonian is $n^{2}$ times a linear combination of the homomorphism densities of chosen finite graphs, one of which is the graph $K_{2}$ with one edge. The coefficients of the combination are the parameters of the model. The GHS inequality, which Griffiths, Hurst and Sherman proved for Ising ferromagnets with pair interactions in a nonnegative field, states for an exponential random graph that the third joint cumulant of any three edge indicators is nonpositive. Bianchi, Collet and Magnanini proved the GHS inequality in a region of the parameters of the two-star model, whose chosen graphs are $K_{2}$ and a path with two edges. For a general exponential random graph whose parameters other than that of $K_{2}$ are nonnegative, they showed that the inequality fails when the probability of an edge, which is the same for all edges, is below $1/2$, and conjectured that it holds whenever that probability is at least $1/2$. We show that, whenever a chosen graph H has at least three edges, the conjecture fails even when every chosen graph other than $K_{2}$ and H has parameter 0. When the chosen graphs are $K_{2}$ and H, and n is at least the number of vertices of H, the probability of an edge is at least $1/2$ at every nonnegative parameter of $K_{2}$ and every positive parameter of H, whereas the inequality fails at every nonnegative parameter of $K_{2}$ when the parameter of H is positive and below a bound that depends on n and on the parameter of $K_{2}$. There is also an explicit bound, which does not depend on n, such that, at every positive parameter of H below it, the inequality fails for every large n at some parameter of $K_{2}$ at which the probability of an edge exceeds $1/2$. When every chosen graph has at most two edges and the parameters other than that of $K_{2}$ are nonnegative, the model is an Ising ferromagnet with pair interactions in a uniform field that is nonnegative exactly when the probability of an edge is at least $1/2$, so the conjecture holds by the inequality of Griffiths, Hurst and Sherman. For the two-star model with a nonnegative parameter of the path, that field is a linear function of the two parameters, so the GHS inequality holds exactly on one side of an explicit line.