  Recently, a new model extending the standard replicator equation to a finite
set of players connected on an arbitrary graph was developed in evolutionary
game dynamics. The players are interpreted as subpopulations of
multipopulations dynamical game and represented as vertices of the graph, and
an edge constitutes the relation among the subpopulations. At each instant,
members of connected vertices of the graph play a 2-player game and collect a
payoff that determines if the chosen strategies will vanish or flourish. The
model describes the game dynamics of a finite set of players connected by a
graph emulating the replicator dynamics. It was proved a relation between the
stability of the mixed equilibrium with the topology of the network. More
specifically, the eigenvalues of the Jacobian matrix of the system evaluated at
the mixed steady state are the eigenvalues of the graph's adjacency matrix
multiplied by a scalar. This paper studies the pure (strict) Nash equilibria of
these games and how it connects to the network. We present necessary and
sufficient conditions for a pure steady-state in coordination or
anti-coordination game to be a (strict) Nash Equilibrium.
