On the properties of the exchange graph of a cluster algebra

Abstract

We prove a conjecture about the vertices and edges of the exchange graph of a cluster algebra script A in two cases: when script A is of geometric type and when script A is arbitrary and its exchange matrix is nondegenerate. In the second case we also prove that the exchange graph does not depend on the coefficients of script A. Both conjectures were formulated recently by Fomin and Zelevinsky. © International Press 2008.