Let $G=(V,E)$ be a finite, simple, undirected graph. For each $v\in V$ we define the *path sequence* $\text{ps}^G\_v: \omega\to \omega$ where $\text{ps}\_v(n)$ is the number of vertices that can be reached from $v$ via a [path](https://en.wikipedia.org/wiki/Path_(graph_theory)) of length at most $n$. (So $\text{ps}^G\_v(0) = 1$ for all $v\in V$.) Of course, these path sequences are eventually constant.


What is an example of pairwise non-isomorphic graphs $G\_i = (V\_i, E\_i)$ for $i \in \{1,2\}$ such that there is a bijection $\varphi:V\_i \to V\_2$ and $$\text{ps}^{G\_1}\_v = \text{ps}^{G\_2}\_{\varphi(v)}$$ for all $v\in V$?



Path sequences and isomorphism of graphs
Take two nonisomorphic $k$-regular graphs of order $n$ and diameter $2$. For any vertex $v$ in such a graph, $|\{w:d(v,w)\le r\}|$ is $1$ if $r=0$, $k+1$ if $r=1$, and $n$ if $r\gt1$.


With $k=3$ and $n=6$ you can take $K\_{3,3}$ and the [triangular prism graph](https://en.wikipedia.org/wiki/Prism_graph). They are nonisomorphic because only one of them has a connected complement, and only one of them is planar.


