  We analyze the topological properties of a family of generalized
Su-Schrieffer-Heeger (SSH) chains and mesh geometries. In both the geometries
the usual staggering in the distribution of the two overlap integrals is
delayed (in space) by the inclusion of a third (additional) hopping term. A
tight-binding Hamiltonian is used to unravel the topological phases,
characterized by a topological invariant. While in the linear chains, the
topological invariant (the Zak phase) always appears to be quantized, in the
quasi-one dimensional strip geometries and the generalized SSH mesh patterns
the quantization of the Zak phase is sensitive to the strength of the
additional interaction (the `extra' hopping integral). We study its influence
thoroughly and explore the edge states and their robustness against disorder in
the cross-linked generalized SSH mesh geometries. The systems considered here
can be taken to model (though crudely) two-dimensional polymers where the
cross-linking brings in non-trivial modification of the energy bands and
transport properties. In addition to the topological features studied, we
provide a prescription to unravel any flat, non-dispersive energy bands in the
mesh geometries, along with the structure and distribution of the compact
localized eigenstates. Our results are analytically exact.
