I need to determine the spectrum of the complete p-partite graph ( in which each partite set has m vertices) using the complement. How can i show this?
I know the spectrum of the adjacency matrix of the complement is given by J-I-A(G) where J is the matrix of ones, I is the identity and A(G) is the adjacency matrix of G. How can I continue?
Things to keep in mind:
The complement of a $p$-partite graph with every part having $m$ vertices is a disjoint union of $p$ copies of $K_m$.
The spectrum of a graph is the union of the spectra of its connected components.
The spectrum of $K_m$ is $m-1$ (multiplicity $1$) and $-1$ (multiplicity $m-1$).
If $H$ is a $k$-regular graph on $n$ vertices with spectrum $k\geq \lambda_2\geq \cdots\geq \lambda_n$, then the spectrum of the complement $\overline{H}$ is $n-k-1 \geq -1-\lambda_n \geq\cdots \geq -1-\lambda_2$