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I would like to know if there is any class of graphs for which the adjacency matrices are invertible. At this moment I am aware of only the class of graphs $n K_2$ which is the disjoint union of $n$ number of $K_2$ matrices where the adjacency matrices are self-inverse.

Are there any other classes?

Omar Shehab
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    Related: http://math.stackexchange.com/questions/1188600/on-the-invertibility-of-adjacency-matrix – Bach Nov 12 '15 at 10:43

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Given a permutation $\pi$ of a finite set $V$, form its cycle graph $G$ as follows: the vertex set is $V$ and the edges are pairs $(v,w)$ for which $\pi(v)=w$. (This is a simple directed graph.) The adjacency matrix will in fact be the permutation matrix corresponding to $\pi$, which is invertible.

We can also form graphs with loops whose adjacency matrices are upper triangular: take the vertex set $\{1,\cdots,n\}$ and adjoin edges $i\to j$ as one wishes but only when $i\le j$ (and of course make sure every vertex has at least one loop).

anon
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  • Are there any other classes? – Omar Shehab Oct 07 '15 at 17:30
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    @OmarShehab Yes, for instance $[\begin{smallmatrix}2 & 1 \ 1 & 1 \end{smallmatrix}]$. A complete classification would seem to characterize all invertible matrices with nonnegative integer entries, and I am not aware of such a characterization (although I haven't looked). My guess is it'd be unlikely there is a strictly graph-theoretic characterization of these graphs, but you'd have to ask a graph theorist to be more confident. – anon Oct 07 '15 at 17:39
  • Just asked a supplementary question at http://cstheory.stackexchange.com/questions/32743/graph-isomorphism-problem-with-invertible-adjacency-matrices – Omar Shehab Oct 07 '15 at 17:41
  • For the first class of graphs, the adjacency matrices are always involutory, right? – Omar Shehab Oct 07 '15 at 20:50
  • I am not sure what definition of 'cycle graph' you are using here. According the the definition in Wikipedia (https://en.wikipedia.org/wiki/Cycle_graph), a cycle graph can have only one cycle. In that case, there cannot be any cycle graph for the permutation $(1 2)(3 4)$, right? – Omar Shehab Oct 07 '15 at 21:09
  • Is there a standard name for the first class of graphs? – Omar Shehab Oct 07 '15 at 21:21
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    @OmarShehab I don't know what the term in graph theory would be or if there is one (it would seem the term "cycle graph" that graph-theorists use is a more restrictive notation), but I'm almost certain I have seen the term "cycle graph of a permutation" used for this construction in the theory of combinatorial species (a mixture of combinatorics, permutation group theory and category theory). – anon Oct 07 '15 at 23:36
  • So, for your definition, a cycle graph is a disjoint collection of cycles, right? – Omar Shehab Oct 08 '15 at 03:12
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    The cycle graph of a permutation, specifically, is what I defined in my answer, yes. – anon Oct 08 '15 at 03:18