Such graphs may exist but it is difficult to characterize them

Yes you are right. But we can frame the above question as follows.

In other words we can say integral graphs whose spectral radius is the maximum degree and least eigenvalue is the minimum degree of a graph.

For such extremums, I suggest you follow Article Bounds on the largest of minimum degree eigenvalues of graphs

https://www.jstor.org/stable/24896701?seq=1

https://math.vanderbilt.edu/ellingmn/paper/srgs/afinal-p.pdf

Dear Ali,

Thank you for providing the reference papers.

Can start with graphs having mimum distinct eigenvalues. Even search in some available spectrum of some good number of graphs, like Cvetcovic and Sachs book.

For regular graph the maximum eigenvalues is its degree. About minimum eigenvalues, then we may consider absolute value of least eigenvalue (because graph has both positive and negative eigenvalues). If this then Regular bipartite graph id the example. Non-regular need to see.