In graph theory, a walk-regular graph is a simple graph where the number of closed walks of any length from a vertex to itself does only depend on but not depend on the choice of vertex. Walk-regular graphs can be thought of as a spectral graph theory analogue of vertex-transitive graphs. While a walk-regular graph is not necessarily very symmetric, all its vertices still behave identically with respect to the graph's spectral properties.
Equivalent definitions Suppose that is a simple graph. Let denote the adjacency matrix of , denote the set of vertices of , and denote the characteristic polynomial of the vertex-deleted subgraph for all Then the following are equivalent: is walk-regular. is a constant-diagonal matrix for all for all
Examples The vertex-transitive 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.
Properties A walk-regular graph is necessarily a regular graph, since the number of closed walks of length two starting at any vertex is twice the vertex's degree. 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 -walk-regular if for any two vertices and of distance at most the number of walks of length from to depends only on and .
The class of -walk-regular graphs is exactly the class of walk-regular graphs
In analogy to walk-regular graphs generalizing vertex-transitive graphs, 1-walk-regular graphs can be thought of as generalizing symmetric graphs, that is, graphs that are both vertex- and edge-transitive. For example, the Hoffman graph is 1-walk-regular but not symmetric.
If is at least the diameter of the graph, then the -walk-regular graphs coincide with the distance-regular graphs. In fact, if and the graph has an eigenvalue of multiplicity at most (except for eigenvalues and , where is the degree of the graph), then the graph is already distance-regular.
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