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Let $G$ be a finite directed acyclic graph. The Cartan matrix $C_G=C$ of $G$ is defined as the matrix with rows and colums indexed by the vertices of $G$ and $c_{i,j}$ counts the number of paths from $...

On the wikipedia page for the Kirchoff matrix theorem, they state a souped up version of the theorem: Let $G$ be a finite undirected loopless graph and let us form the square matrix $L$ indexed by the ...