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Oct 23, 2010 at 13:53 comment added Qiaochu Yuan @Seva: you can consider graphs with loops. I have never seen the point of ignoring loops or multiple edges for the purposes of analyzing matrices like M_n.
Oct 23, 2010 at 11:22 comment added J. M. isn't a mathematician He doesn't have the rep to post a comment, though.
Oct 23, 2010 at 11:16 comment added Seva There are no graphs for which $M_n$ are adjacency matrices since each $M_n$ has non-zero elements on the main diagonal. You can, of course, consider the graph on the vertex set $[n]$ with two distinct vertices $i,j\in[n]$ adjacent if and only if $ij\le n$, and attach loops to those vertices $i\in[n]$ with $i^2\le n$. Anyway, this does not help much. An editing request: would you mind to re-post your question as a comment to the problem, not as an answer?
Oct 23, 2010 at 10:49 history answered KexiangXu CC BY-SA 2.5