Graphs with (k,τ)-regular sets and equitable partitions are examples of graphs with regularity constraints. A (k,τ)-regular set of a graph G is a subset of vertices S⊆V(G) inducing a k-regular subgraph and such that each vertex not in S has τ neighbors in S. The existence of such structures in a graph provides some information about the eigenvalues and eigenvectors of its adjacency matrix. For example, if a graph G has a (k 1 ,τ 1 )-regular set S 1 and a (k 2 ,τ 2 )-regular set S 2 such that k 1 −τ 1 =k 2 −τ 2 =λ, then λ is an eigenvalue of G with a certain eigenvector. Additionally, considering primitive strongly regular graphs, a necessary and sufficient condition for a particular subset of vertices to be (k,τ)-regular is introduced. Another example comes from the existence of an equitable partition in a graph. If a graph G, has an equitable partition π then its line graph, L(G), also has an equitable partition, π¯, induced by π, and the adjacency matrix of the quotient graph L(G)/π¯ is obtained from the adjacency matrix of G/π.