How much symplectic geometry can be done on finite simple graphs? Is there an analogue of the Arnold conjecture on graphs bounding the number of periodic solutions of a symplectic Hamiltonian map from below by the sum of the Betti numbers? This 25 minute presentation is an attempt to organize my thoughts about this. Cartan's calculus allows to define notions of vector fields on graphs in terms of an inner derivative i_X which is the graph case just a matrix satisfying i_X^2=0 leading to a Lie derivative L_X = i_X d + d i_X. These vector fields form a Lie algebra. In the symplectic case, if an exact, nowhere vanishing 2-form w is given on the graph, one can look at vector fields satisfying i_X w = dH, a relation again borrowed from the continuum. These symplectic fields form a sub Lie algebra. Solutions of the Wave equation can be seen as discrete analogues of solutions of the Hamiltonian system defined by the Hamiltonian H. The Hodge Laplacian L=d d^* + d^* d has b = sum_k b_k zero eigenvalues. In the continuum, this is a lower bound on the number of fixed points of a Hamiltonian symplectomorphism. It is very natural to ask what happens with the Harmonic forms when L is changed to L_X. An Arnold type question is to estimate the number of periodic solutions of the wave equation defined by L_X.

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