Additive functions on quivers Original Research Article

An integral function on the set of vertices of a graph is additive if twice is value at any vertex v equals the sum of its values at all adjacent vertices, counting multiple edges. It is well known that among finite connected graphs exactly the extended Dynkin graphs admit a positive additive function, whereas the Dynkin diagrams themselves only allow almost-additive functions, violating additivity in a single vertex. In the present paper we study—usually non-positive—additive or non-additive functions on finite quivers, and relate the concept of additivity to the radical of the homological Euler form. Our main results concern the existence and construction of such functions for wild quivers. Our results are most specific in case the underlying graph is a tree, possibly with multiple edges