  We present a simple criterion for solvability of lattice spin systems on the
basis of the graph theory and the simplicial homology. The lattice systems
satisfy algebras with graphical representations. It is shown that the null
spaces of adjacency matrices of the graphs provide conserved quantities of the
systems. Furthermore, when the graphs belong to a class of simplicial
complexes, the Hamiltonians are found to be mapped to bilinear forms of
Majorana fermions, from which the full spectra of the systems are obtained. In
the latter situation, we find a relation between conserved quantities and the
first homology group of the graph, and the relation enables us to interpret the
conserved quantities as flux excitations of the systems. The validity of our
theory is confirmed in several known solvable spin systems including the 1d
transverse-field Ising chain, the 2d Kitaev honeycomb model and the 3d diamond
lattice model. We also present new solvable models on a 1d tri-junction, 2d and
3d fractal lattices, and the 3d cubic lattice.
