# When does the adjacency matrix of a graph have an eigenvalue zero?

When does the adjacency matrix $A$ of an undirected graph $G$ not have full rank? Is there any intuition to understanding this?

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It can be a little tricky. There was a question here a few days ago about cycle graphs --- just $n$ vertices in a single cycle, $n\ge3$. Turns out in this case that the adjacency matrix has full rank if and only if $n$ is not a multiple of $4$. –  Gerry Myerson Nov 19 '12 at 2:35