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I was thinking about adjacency matrices, when I realised that an adjacency matrix is nilpotent if and only if there is no cycle in the graph.

So, I started wondering: if a graph has an adjacency matrix that is invertible, what does it mean for the graph?

I usually define a graph the following way (but if the is some proprieties with another definition, I'm also interested): a graph $G=(V,E)$ is couple of set where $V = [\![1;n]\!]$ and $E \subset \{ (i,j) \in V^2, i\neq j$}. The adjacency matrix $M=(M_{ij})$ is defined such as $M_{ij} = 1$ if $(i,j) \in V$, $0$ otherwise.

Here I'm talking about undirected graphs.

MiKiDe
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  • Just to make sure, you're talking about directed graphs specifically here? The undirected case is asked already at at https://math.stackexchange.com/questions/1188600/on-the-invertibility-of-the-adjacency-matrix-of-a-graph – hmakholm left over Monica Sep 08 '18 at 17:25
  • Yes I'm talking about undirected graphs. I'll make it clear in my post! – MiKiDe Sep 09 '18 at 08:34

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