As a generalization of orbit-polynomial and distance-regular graphs, we introduce the concept of a quotient-polynomial graph. In these graphs every vertex uinduces 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.