Hall algebras of graphs and rooted trees

Abstract

Hall algebras can be associated with a broad class of combinatorial structures through the theory of 2-Segal sets. In this paper, we study the Hall algebras arising from the 2-Segal sets of rooted trees, undirected graphs, and directed graphs. In each case, we establish an analogue of Green's theorem giving a twisted bialgebra structure to the Hall algebra, describe the primitive elements, and derive a presentation by generators and relations. As an application, we realize the Hall algebra of an undirected graph as the cohomology ring of a topological space. In the case of directed graphs, we introduce the Hall polynomial, defined as the Poincar{é} polynomial of the space of primitive elements. The Hall polynomial is an invariant of the underlying undirected graph which we show to be closely related to the Tutte polynomial. We furthermore give examples of graphs with equal Tutte polynomials but distinct Hall polynomials showing thus that the Hall polynomial encodes different information from that of the Tutte polynomial.