Let $G$ be a finite connected graph. Let $K$ be a 2-dimensional complex such that $K^{(1)} = G$, $\tilde{H}_2(K)=0$ and $\tilde{H}_1(K)=\Bbb Z_m$. Show that $\det\partial_2^t∂_2 =m^2 ·k(G).$
Over the above statement, the $\tilde{H}$ notates the reduced simplicial homology and $k(G)$ notates the number of spanning trees of given graph G.
Where to start to prove the square of boundary operator results to be same as $m^2\cdot k(G)$?