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Determinantshard
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According to the Matrix Tree Theorem, the number of spanning trees of a graph is equal to any cofactor of its Laplacian matrix LLL. For a cycle graph C4C_4C4​ on 4 vertices, the Laplacian matrix is: L=(2−10−1−12−100−12−1−10−12)L = \begin{pmatrix} 2 & -1 & 0 & -1 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ -1 & 0 & -1 & 2 \end{pmatrix}L=​2−10−1​−12−10​0−12−1​−10−12​​ What is the cofactor C11C_{11}C11​ of LLL, which represents the number of spanning trees of C4C_4C4​?