# Eigenvectors of asymmetric graphs

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!

• By asymmetric, you mean the automorphism group is trivial? Aug 8 '12 at 18:44
• Yes, the graph has a trivial automorphism group. And yes I'm assuming a connected graph. I edited the question to reflect this. Aug 9 '12 at 6:23
• I'm curious what was the motivation for the question. Aug 9 '12 at 12:42

I think the answer is no. Take the Frucht Graph, the simplest nontrivial asymmetric graph. Its adjacency matrix is

\begin{equation*} \left( \begin{array}{cccccccccccc} 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\\\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\\\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \end{array} \right) \end{equation*}

none of whose eigenvectors seems to have distinct entries.

• If I entered it correctly into Maple then three of the 12 distinct eigenvalues do have eigenvectors with distinct entries. Aug 9 '12 at 2:21
• I cut and pasted the above matrix into Mathematica and it seems that all eigenvectors have repeated entries. Aug 9 '12 at 3:06
• Ok, I agree.I think I got fooled when Maple took equal but not identically written algebraic integers and evaluated them slightly differently numerically. Aug 9 '12 at 4:55
• Funny that I had checked the Frucht graph before asking this question and hadn't noticed that all eigenvectors' entries were not distinct. Well spotted. Aug 9 '12 at 6:29