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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

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See the paper by Irene Sciriha, A characterization of singular graphs, Electronic Journal of Linear Algebra, Volume 16, pp. 451-462, December 2007, available at a URL I can't post because when I try I get the error message,

Oops! Your answer couldn't be submitted because:

Your post contains a link to the invalid host '212.189.136.211'. Please correct it by specifying a non numeric domain or wrapping it in a code block.

Anyway, it has that numeric domain, then journals/ELA/ela-articles/articles/vol16_pp451-462.pdf

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