Walk-regular graph

In discrete mathematics, a walk-regular graph is a simple graph where the number of closed walks of any length from a vertex to itself does not depend on the choice of vertex.
Equivalent definitions.
Suppose that formula_1 is a simple graph. Let formula_2 denote the adjacency matrix of formula_1, formula_4 denote the set of vertices of formula_1, and formula_6 denote the characteristic polynomial of the vertex-deleted subgraph formula_7 for all formula_8Then the following are equivalent: