Additive functions on trees
The motivation for considering positive additive functions on trees was a characterization of extended Dynkin graphs (see I. Reiten [R]) and applications of additive functions in representation theory (see H. Lenzing and I. Reiten [LR] and T. Hübner [H]).
We consider graphs equipped with integer-valued functions, i.e. valued graphs (see also [DR]). Methods are given for constructing additive functions on valued trees (in particular on Euclidean graphs) and for characterizing their structure. We introduce the concept of almost additive functions, which are additive on each vertex of a graph except one (called the exceptional vertex). On (valued) trees (with fixed exceptional vertex) the almost additive functions are unique up to rational multiples. For valued trees a necessary and sufficient condition is given for the existence of positive almost additive functions.