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If $G = (V, E)$ is a simple, undirected graph and $T \subseteq V$, let $$N(T) = \{v \in V: \{v, t\}\in E \text{ for some }t\in T\}.$$

Given $v\in V$ we let $N_0(v) = \{v\}$ and $N_{k+1}(v) = N_k(v) \cup N(N_k(v))$ for all $k\geq 1$. The iterated degree sequence of $v$, denoted by $(\text{deg}_k(v))_{k\in\omega}$, is defined by $$\text{deg}_k(v) = |N_k(v)|\text{ for every }k\in \omega.$$

To every finite graph $G = (V,E)$ we associate the iterated degree matrix $\mathbb{D}(G) \in \mathbb{N}^{n\times n}$ (where $n=|V|$) in the following way: for every $v\in V$, take the first $n$ elements of its iterated degree sequence; order these $n$-element integer vectors lexicographically, and put these lexicographically ordered vectors in the matrix.

Question. Are there finite $G_i = (V_i, E_i)$ for $i = 1,2$ with $|V_1| = |V_2|$, $G_1\not\cong G_2$, but $\mathbb{D}(G_1) = \mathbb{D}(G_2)$?

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Yes. Consider all graphs with $V=\{1,\ldots,n\}$ for which the vertex $n$ has degree $n-1$. There are $2^{n^2/2+o(n^2)}$ isomorphism classes of such graphs. But $\deg_k(v)=n$ for all $v$ and all $k\geqslant 2$, thus there exist only at most $n^{n-1}$ distinct matrices.

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  • $\begingroup$ I don't understand the construction: take n=2, so V={1,2}, vertex 1 has degree 0, sure, vertex 1 has degree 1... how? Connected to what other vertex? $\endgroup$ Jul 20 at 6:58
  • $\begingroup$ @JacquesCarette we fix only the the degree of vertex $n$, not other vertices. And this all makes sense only for large $n$. $\endgroup$ Jul 20 at 9:57
  • $\begingroup$ Oh, I misunderstood, I thought you meant vertex i has degree i-1 -- this makes much more sense! $\endgroup$ Jul 20 at 20:01

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