  We describe a model for $m$ vertex reinforced interacting random walks on
complete graphs with $d\geq 2$ vertices. The transition probability of a random
walk to a given vertex depends exponentially on the proportion of visits made
by all walks to that vertex. The individual proportion of visits is modulated
by a strength parameter that can be set equal to any real number. This model
covers a large variety of interactions including different vertex repulsion and
attraction strengths between any two random walks as well as self-reinforced
interactions. We show that the process of empirical vertex occupation measures
defined by the interacting random walks converges (a.s.) to the limit set of
the flow induced by a smooth vector field. Further, if the set of equilibria of
the field is formed by isolated points, then the vertex occupation measures
converge (a.s.) to an equilibrium of the field. These facts are shown by means
of the construction of a strict Lyapunov function. We show that if the absolute
value of the interaction strength parameters are smaller than a certain upper
bound, then, for any number of random walks ($m\geq 2$) on any graph ($d \geq
2$), the vertex occupation measure converges toward a unique equilibrium. We
provide two additional examples of repelling random walks for the cases $m=d=2$
and $m=3$, $d=2$. The latter is used to study some properties of three
exponentially repelling random walks on $\mathbb{Z}$.
