  The paper examines a class of first order linear hyperbolic systems, proposed
as a generalization of the Goldstein-Kac model for velocity-jump processes and
determined by a finite number of speeds and corresponding transition rates. It
is shown that the large-time behavior is described by a corresponding scalar
diffusive equation of parabolic type, defined by a diffusion matrix for which
an explicit formula is given. Such representation takes advantage of a variant
of the Kirchoff's matrix tree Theorem applied to the graph associated to the
system and given by considering the velocities as verteces and the transition
rates as weights of the arcs.
