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 $$G$$ is a simple graph. Let $$A$$ denote the adjacency matrix of $$G$$, $$V(G)$$ denote the set of vertices of $$G$$, and $$\Phi_{G - v}(x)$$ denote the characteristic polynomial of the vertex-deleted subgraph $$G - v$$ for all $$v \in V(G).$$Then the following are equivalent:
 * $$G$$ is walk-regular.
 * $$A^k$$ is a constant-diagonal matrix for all $$k \geq 0.$$
 * $$\Phi_{G - u}(x) = \Phi_{G - v}(x)$$ for all $$u, v \in V(G).$$

Examples

 * The vertex-transitive graphs are walk-regular.
 * The semi-symmetric graphs are walk-regular.
 * The distance-regular graphs are walk-regular. More generally, any simple graph in a homogeneous coherent algebra is walk-regular.
 * A connected regular graph is walk-regular if:
 * It has at most four distinct eigenvalues.
 * It is triangle-free and has at most five distinct eigenvalues.
 * It is bipartite and has at most six distinct eigenvalues.

Properties

 * A walk-regular graph is necessarily a regular graph.
 * Complements of walk-regular graphs are walk-regular.
 * Cartesian products of walk-regular graphs are walk-regular.
 * Categorical products of walk-regular graphs are walk-regular.
 * Strong products of walk-regular graphs are walk-regular.
 * In general, the line graph of a walk-regular graph is not walk-regular.

k-walk-regular graphs
A graph is $$k$$-walk regular if for any two vertices $$v$$ and $$w$$ of distance at most $$k,$$ the number of walks of length $$l$$ from $$v$$ to $$w$$ depends only on $$k$$ and $$l$$.

For $$k=0$$ these are exactly the walk-regular graphs.

If $$k$$ is at least the diameter of the graph, then the $$k$$-walk regular graphs coincide with the distance-regular graphs. In fact, if $$k\ge 2$$ and the graph has an eigenvalue of multiplicity at most $$k$$ (except for eigenvalues $$d$$ and $$-d$$, where $$d$$ is the degree of the graph), then the graph is already distance-regular.