Assume we have a complete graph with $N$ nodes. If we show the Laplacian graph with $L$, we know that one eigenvalue of $L$ is equal to zero and all the others are equal to $N$, so we have $N-1$ eigenvalues equal to $N$.
My question is about elements of each eigenvector related to these eigenvalues. Can we say what the eigenvectors are without calculating them and their elements? Actually we have $N$ eigenvectors which the first one has the same elements equal to $\sqrt \frac{1}{N}$. Can we say anything about elements of other $N-1$ eigenvectors?