When does the adjacency matrix $A$ of an undirected graph $G$ not have full rank? Is there any intuition to understanding this?
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,
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Anyway, it has that numeric domain, then journals/ELA/ela-articles/articles/vol16_pp451-462.pdf