  Predictions of localized Majorana modes, and ideas for manipulating these
degrees of freedom, are the two key ingredients in proposals for physical
platforms for Majorana quantum computation. Several proposals envisage a
scalable network of such Majorana modes coupled bilinearly to each other by
quantum-mechanical mixing amplitudes. Here, we develop a theoretical framework
for characterizing collective topologically protected zero-energy Majorana
fermion excitations of such networks of localized Majorana modes. A key
ingredient in our work is the Gallai-Edmonds decomposition of a general graph,
which we use to obtain an alternate ``local'' proof of a ``global'' result of
Lov{\'a}sz and Anderson on the dimension of the topologically protected null
space of {\em real skew-symmetric} (or pure-imaginary hermitean) adjacency
matrices of general graphs. Our approach to Lov{\'a}sz and Anderson's result
constructs a maximally-localized basis for the said null-space from the
Gallai-Edmonds decomposition of the graph. Applied to the graph of the Majorana
network in question, this gives a method for characterizing basis-independent
properties of these collective topologically protected Majorana fermion
excitations, and relating these properties to the correlation function of
monomers in the ensemble of maximum matchings (maximally-packed dimer covers)
of the corresponding network graph. Our approach can also be used to identify
signatures of zero-energy excitations in systems modeled by a free-fermion
Hamiltonian with a hopping matrix of this type; an interesting example is
provided by vacancy-induced Curie tails in generalizations (on non-bipartite
lattices) of Kitaev's honeycomb model.
