  We propose a general scheme to construct a Hamiltonian $H_{\text{root}}$
describing a square root of an original Hamiltonian $H_{\text{original}}$ based
on the graph theory. The square-root Hamiltonian is defined on the subdivided
graph of the original graph of $H_{\text{original}}$, where the subdivided
graph is obtained by putting one vertex on each link in the original graph.
When $H_{\text{original}}$ describes a topological system, there emerge in-gap
edge states at non-zero energy in the spectrum of $H_{\text{root}}$, which are
the inherence of the topological edge states at zero energy in
$H_{\text{original}}$. In this case, $H_{\text{root}}$ describes a square-root
topological insulator or superconductor. Typical examples are square roots of
the Su-Schrieffer-Heeger (SSH) model, the Kitaev topological superconductor
model and the Haldane model. Our scheme is also applicable to non-Hermitian
topological systems, where we study an example of a nonreciprocal non-Hermitian
SSH model.
