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](https://en.wikipedia.org/wiki/Lexicographic_order), and put these lexicographically ordered vectors in the matrix.


**Question.** Are there finite $G\_i = (V\_i, E\_i)$ for $i = 1,2$ with $|V\_1| = |V\_2|$, $\mathbb{D}(G\_1) = \mathbb{D}(G\_2)$, but $\chi(G\_1) \neq \chi(G\_2)$?



Do graphs with identical degree matrix have the same chromatic number?
There are such examples. Take two graphs with the same degrees but different chromatic number (for example, a cycle of length 6 and two triangles). Add a vertex of full degree to both.


