  We present the theory and experimental demonstration of a topological
classification of finite tight binding Hamiltonians with chiral symmetry. Using
the graph-theoretic notion of complete matchings, we show that many chiral
tight binding structures can be divided into a number of sections, each of
which has independent topological phases. Hence the overall classification is
$N\mathbb{Z}_2$, corresponding to $2^N$ distinct phases, where $N$ is the
number of sections with a non-trivial $\mathbb{Z}_2$ classification. In our
classification, distinct topological phases are separated by exact closures in
the energy spectrum of the Hamiltonian, with degenerate pairs of zero energy
states. We show that that these zero energy states have an unusual localisation
across distinct regions of the structure, determined by the manner in which the
sections are connected together. We use this localisation to provide an
experimental demonstration of the validity of the classification, through radio
frequency measurements on a coaxial cable network which maps onto a tight
binding system. The structure we investigate is a cable analogue of an ideal
graphene ribbon, which divides into four sections and has a $4\mathbb{Z}_2$
topological classification.
