  We generalize a result from Volkov [Ann. Probab. 29 (2001) 66--91] and prove
that, on a large class of locally finite connected graphs of bounded degree
$(G,\sim)$ and symmetric reinforcement matrices $a=(a_{i,j})_{i,j\in G}$, the
vertex-reinforced random walk (VRRW) eventually localizes with positive
probability on subsets which consist of a complete $d$-partite subgraph with
possible loops plus its outer boundary. We first show that, in general, any
stable equilibrium of a linear symmetric replicator dynamics with positive
payoffs on a graph $G$ satisfies the property that its support is a complete
$d$-partite subgraph of $G$ with possible loops, for some $d\ge1$. This result
is used here for the study of VRRWs, but also applies to other contexts such as
evolutionary models in population genetics and game theory. Next we generalize
the result of Pemantle [Probab. Theory Related Fields 92 (1992) 117--136] and
Bena\"{{\i}}m [Ann. Probab. 25 (1997) 361--392] relating the asymptotic
behavior of the VRRW to replicator dynamics. This enables us to conclude that,
given any neighborhood of a strictly stable equilibrium with support $S$, the
following event occurs with positive probability: the walk localizes on
$S\cup\partial S$ (where $\partial S$ is the outer boundary of $S$) and the
density of occupation of the VRRW converges, with polynomial rate, to a
strictly stable equilibrium in this neighborhood.
