  In this article we show that there are branching systems (which induce
representations of the graph algebra $C^*(E)$) associated to each row-countable
graph $E$. For row-countable graphs, we characterize the condition $(L)$ via
branching systems. Moreover, we show that each permutative representation in
Hilbert spaces operators is unitarily equivalent to one induced by a branching
system, even the spaces being not separable. Furthermore, under some hypothesis
on the graph, we show that each representation of the graph C*-algebra is
permutative.
