  In this paper, we introduce a new method to compute magnitude homology of
general graphs. To each direct sum component of magnitude chain complexes, we
assign a pair of simplicial complexes whose simplicial chain complex is
isomorphic to it. First we states our main theorem specialized to trees, which
gives another proof for the known fact that trees are diagonal. After that, we
consider general graphs, which may have cycles. We also demonstrate some
computation as an application.
