  As a generalization of orbit-polynomial and distance-regular graphs, we
introduce the concept of a quotient-polynomial graph. In these graphs every
vertex $u$ induces the same regular partition around $u$, where all vertices of
each cell are equidistant from $u$. Some properties and characterizations of
such graphs are studied. For instance, all quotient-polynomial graphs are
walk-regular and distance-polynomial. Also, we show that every
quotient-polynomial graph generates a (symmetric) association scheme.
