Eigen value and Regular Graph (not Strongly Regular graph). $A,B$ are 2 adjacency matrices of $d$ Regular graphs(not Strongly Regular graphs). I would like to know- 1.Results/ information related to Eigen values of A,B. There is a formula for strongly Regular graph's eigen value and there are exactly 3 eigen values. Is there something for Regular graphs analogous to strongly Regular graph's ?? 2\. When it is guaranteed that $A,B$ will have at least one different eigen value ??

A regular graph has its valency as its largest eigenvalue, if the graph is not regular, the largest eigenvalue is less than the valency. But this is practically the only thing that distinguishes the spectral theory of regular graphs from that of general graphs. In particular there is no formula of any sorts for the eigenvalues of regular graphs in general.

As for general graphs, there is is no useful characterization of the regular graphs that are determined by their spectrum.