Such graphs are studied a lot in coding theory and in the theory of association schemes. In paricular the eigenvalues can be explicitly written down  in terms of Krawchuk polynomials. 

You can find this e.g. in Bannai  & Ito "Association Schemes I": https://www.amazon.com/Algebraic-Combinatorics-Association-Schemes-Mathematics/dp/0805304908

I never saw inertia of these graphs mentioned anywhere, but it looks easy to figure out.