  In this paper, we deal with a notion of Banach space-valued mappings defined
on a set consisting of finite graphs with uniformly bounded vertex degree.
These functions will be endowed with certain boundedness and additivity
criteria. We examine their normalized long-term behaviour along a particular
class of graph sequences. Using techniques developed by Elek, we show
convergence in the topology of the Banach space if the corresponding graph
sequence possesses a hyperfinite structure. These considerations extend and
complement the corresponding results for amenable groups. As an application, we
verify the uniform approximation of the integrated density of states for
bounded, finite range operators on discrete structures. Further, we extend
results concerning an abstract version of Fekete's Lemma for amenable groups
and cancellative semigroups to the geometric situation of convergent graph
sequences.
