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Take a simple $k-$regular graph (every vertex has degree $k$) $G$ and its adyacency matrix $A$. Then it's known that $k$ is one eigenvalue of $A$ with associated eigenvector $u=\begin{bmatrix}1 & 1 & \cdots & 1\end{bmatrix}^T$. Let $\lambda_2,\dots,\lambda_{n}$ be the remaining eigenvalues.

I want to prove that the complement $\overline{G}$ has the same eigenvectors, with corresponding eigenvalues $n-k-1$, $-1-\lambda_2,\dots,-1-\lambda_{n}$.

Since the adyacency matrix of $\overline{G}$ is $\overline{A}=M-I-A$ where $M$ is a matrix of $1$s, then for an eigenvector $u$ of $A$ with eigenvalue $\lambda$, we have $$\overline{A}u=Mu-u-Au=Mu-u-\lambda u=Mu+(-1-\lambda)u.$$

For the eigenvector $u=\begin{bmatrix}1 & 1 & \cdots & 1\end{bmatrix}^T$, we have $\lambda=k$ and $Mu=nu$ so in this case $\overline{A}u=(n-1-k)u$, so $u$ is also an eigenvector of $\overline{A}$.

For the remaining eigenvectors $u$, to prove the result I need to prove that $Mu=0,$ i.e. the sum of the components of $u$ is $0$. How can I prove that?

By the way, I found a result in Dragos book about Graph Spectra, in which they prove the characteristic polynomial $P_{\overline{G}}$ of $\overline{G}$ is $(-1)^n\frac{x-n+r+1}{x+r+1} P_G(-x-1)$, and in the proof they use the fact that $Mu=0$ for every but one eigenvector (The vector of $1$s) without any details, so I think I must be missing a simple argument for that.

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The adjacency matrix is symmetric, so it has an orthonormal basis of eigenvectors. The first of these eigenvectors, a multiple of $[1\;1\;\cdots\;1]$, is orthogonal to all others, which is just the condition that the entries of any other eigenvector must sum to $0$.

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  • $\begingroup$ That's just it. Thanks. $\endgroup$ – David Molano Jan 27 at 21:34

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