Let $G$ be an asymmetric connected graph. Then is it always the case that at least one of the eigenvectors of its adjacency matrix $A$ consists entirely of distinct entries? Thanks!
This question is motivated by a talk I went to earlier today. Suppose we have a $d$-regular graph $G$ with $n$ vertices, with adjacency matrix $A$. Let $$\lambda_1\geq \lambda_2 \geq\dots \geq ...