We now introduce equitable partitions, which enable us to obtain information about eigenvalues and eigenvectors of a graph from a smaller ‘quotient’. We map out the basic theory in Sections 1 and 2, and in Section 3 we apply it to walk-regular graphs, which may be viewed as a combinatorial generalisation of vertex-transitive graphs. Section 4 introduces Haemer’s theory of generalised interlacing. We use this to derive a lower bound on the chromatic number of a graph due to A. Hoffman. The final two sections introduce covering graphs and present a bound on the spectral radius of a tree. The latter will prove useful in next chapter.