Is a simple graph connected, if every node has at least one adjacent edge and $|E|\ge |V|-1$?
If $|V|=n, |E|\ge n-1$ and  every node has at least one adjacent edge , then $G$ is connected.


No, it is not true. Here is an counterexample.



Graph made at https://graphonline.ru/en/

Exercise. Construct a simpler counterexample where $|V|=n$ and $|E|=n-1$.