Abstract
The motivation of considering positive additive functions on trees was the characterization of extended Dynkin graphs (see I. Reiten [R]) and the application of additive functions in the representation theory (see H. Lenzing and I. Reiten [LR] and T. H¨ubner [H]). We consider graphs equipped with functions of integer values, i.e.valued graphs (see also [DR]). Methods are given for the construction of additive functions on valued trees (in particular on Euclidean graphs) and for the characterization of their structure. We introduce the concept of almost additive functions, which are additive on each vertex of a graph except for one (called 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.
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