Suppose $G, H$ are finite, simple, undirected graphs and there is a bijection between the vertex sets $\varphi:V(G) \to V(H)$ such that for all $v\in V$ we have $$\text{deg}\_G(v) = \deg\_H(\varphi(v)).$$


Does this imply that $G\cong H$? In case the answer is positive, does it remain so if we consider infinite graphs?



Reconstructing a graph from the multiset of degrees
No, consider the [Petersen graph](https://en.wikipedia.org/wiki/Petersen_graph) and another with no star in the middle but just a simple pentagon. The former has no 4-cycle, but the latter has one.


