Necessities for two undirected graphs being isomorphic

Question

As far as I know, for two undirected graphs $G = (V, E) $ and $H = (V', E')$, the following criteria is necessary for them to be isomorphic:

1. $|V| = |V'|$
2. $|E| = |E'|$
3. $G$ has $j$ nodes of degree $k$ $\Leftrightarrow $ $H$ has $j$ nodes of degree $k$

Are there more that are that obvious? Thanks in advance for your help!

Answer 1

For two graphs $G$ and $H$ it is necessary for literally every property that holds of $G$ to also hold of $H$ and vice-versa. There's no point making a list because everything is in that list.