I am looking for a large family (infinite pairs) of cospectral graphs with these condtions:

• The graphs are non-regular,

• Minimum degree is greater than $1$,

• The degree sequences of these cospectral graphs are the same.

I need cospectrality by adjacency matrix and the graphs are simple.

The motivation for asking this question is that; if we have counterexample for reconstruction conjecture, the two graphs have these properties. I want to study such family of graphs.