I didn't see this asked anywhere else, and I'm really struggling with understanding and solving this question:
Let $G$ be a simple k-regular graph.
Show that if $\lambda$ is an eigenvalue of G distinct from $k$, then $-1-\lambda$ is an eigenvalue of $G^C$ with the same multiplicity.
What I know:
Since $G$ is k-regular, it has an eigenvalue of $k$ and corresponding eigenvector 1. Deduced from this is that $G^C$ has eigenvalue $n-k-1$ with corresponding eigenvector 1. I also know that eigenvectors corresponding to distinct eigenvalues of a real symmetric matrix are orthogonal, but I'm not sure if this is helpful.
Thanks in advance. Any help is deeply appreciated.