Let the maximum and minimum degress of a graph be denoted (as usual) by $\Delta$ and $\delta$ respectively.

> 
> A graph is ***almost regular*** if $\Delta-\delta=1$.
> 
> 
> 

Now, here is a simple way to generate such graphs: start with a regular graph and delete a matching. Either that or add a matching. 

> 
> An almost regular graph which is produced from a regular graph by the
>  addition or removal of a matching is ***obvious***.
> 
> 
> 

Can you find examples of non-obvious almost regular graphs? So far, I am unable to produce any but I have a feeling there ought to be some.

A follow-up question, in case non-obvious ones do exist, would of course be to estimate which case is more prevalent.