What are the rights to the father if the mother just walked out and gave children to father ?

Go to family court and apply for full custody and child support.
Even if you do not know where she is they might be able to track her down and go after her for child support. Even if they can't it will be proof in your favor that she left them.

It will depend on the situation. I would suggest you first look at the following link and see what the state statutes say in regards to your situation.

State Statutes

Need more info to give you specifics...

Ok... after looking around a bit... there are no references (that I could find, anyway) to "abandonment" in Nebraska statutes. Doesn't mean it doesn't happen and isn't addressed...

What exactly do you mean by "mother just walked out and gave children to father?" Is dad listed on the birth certificate or just acknowleged as the presumptive father? Were they ever married? Have they ever gone to court previously in regards to custody and support? How long has it been since the mother left the kids with the father and is there an indication that this was a permanent "drop off?" What rights, so-to-speak, are you asking about?