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 ${\displaystyle G}$ is a simple graph. Let ${\displaystyle A}$ denote the adjacency matrix of ${\displaystyle G}$, ${\displaystyle V(G)}$ denote the set of vertices of ${\displaystyle G}$, and ${\displaystyle \Phi _{G-v}(x)}$ denote the characteristic polynomial of the vertex-deleted subgraph ${\displaystyle G-v}$ for all ${\displaystyle v\in V(G).}$Then the following are equivalent:

• ${\displaystyle G}$ is walk-regular.
• ${\displaystyle A^{k}}$ is a constant-diagonal matrix for all ${\displaystyle k\geq 0.}$
• ${\displaystyle \Phi _{G-u}(x)=\Phi _{G-v}(x)}$ for all ${\displaystyle u,v\in V(G).}$

Examples

• The vertex-transitive graphs are walk-regular.
• The semi-symmetric graphs are walk-regular.[1]
• 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.

References

1. "Are there only finitely many distinct cubic walk-regular graphs that are neither vertex-transitive nor distance-regular?". mathoverflow.net. Retrieved 2017-07-21.

External links

• Chris Godsil and Brendan McKay, Feasibility conditions for the existence of walk-regular graphs.