Walk-regular graph

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 ${\displaystyle u,v\in V(G).}$

• 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.

• 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.

A graph is ${\displaystyle k}$-walk regular if for any two vertices ${\displaystyle v}$ and ${\displaystyle w}$ of graph-distance ${\displaystyle \mathrm {dist} (v,w)\leq k}$ the number of walks of length ${\displaystyle \ell }$ from ${\displaystyle v}$ to ${\displaystyle w}$ depends only of ${\displaystyle k}$ and ${\displaystyle \ell }$. [2]

For ${\displaystyle k=0}$ these are exactly the walk-regular graphs.

If ${\displaystyle k}$ is at least the diameter of the graph, then the ${\displaystyle k}$-walk regular graphs coincide with the distance-regular graphs. In fact, if ${\displaystyle k\geq 2}$ and the graph has an eigenvalue of multiplicity at most ${\displaystyle k}$ (except for eigenvalues ${\displaystyle d}$ and ${\displaystyle -d}$, where ${\displaystyle d}$ is the degree of the graph), then the graph is already distance-regular. [3]

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