We study zero modes of Laplacians on compact and non-compact metric graphs with general self-adjoint vertex conditions. In the first part of the paper, the number of zero modes is expressed in terms of the trace of a unitary matrix that encodes the vertex conditions imposed on functions in the domain of the Laplacian. In the second part, a Dirac operator is defined whose square is related to the Laplacian. To accommodate Laplacians with negative eigenvalues, it is necessary to define the Dirac operator on a suitable KreAn space. We demonstrate that an arbitrary, self-adjoint quantum graph Laplacian admits a factorisation into momentum-like operators in a KreAn-space setting. As a consequence, we establish an index theorem for the associated Dirac operator and prove that the zero-mode contribution in the trace formula for the Laplacian can be expressed in terms of the index of the Dirac operator.
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