Merely looking at adjacency matrix of a regular graph, without explicit calculation, can we decide that graph will have an odd eigenvalue or not?
If regularity is odd, we are sure that it will be an odd e.v for graph but when regularity is not odd, what does a graph suggest by an odd e.v.?
Does having odd e.v. has something to do with structure of graph?
Any help or reference is highly appreciated. Thank you.