  We present the exact analytical expression for the spectrum of a sparse
non-Hermitian random matrix ensemble, generalizing two classical results in
random-matrix theory: this analytical expression forms a non-Hermitian version
of the Kesten-Mckay law as well as a sparse realization of Girko's elliptic
law. Our exact result opens new perspectives in the study of several physical
problems modelled on sparse random graphs. In this context, we show
analytically that the convergence rate of a transport process on a very sparse
graph depends upon the degree of symmetry of the edges in a non-monotonous way.
