How ubiquitous are dragon segments in quantum transmission?

Quantum dragon segments are nanodevices that have energy-independent total transmission of electrons. At the level of the single-band tight-binding model a nanodevice is viewed as a weighted undirected graph, with a vertex weight given by the on-site energy and the edge weight given by the tight-binding hopping parameter. A quantum dragon is a weighted undirected graph which when connected to idealized semi-infinite input and output leads, has the electron transmission probability ${\cal T}(E)$$=$$1$ for all electron energies $E$. The probability ${\cal T}(E)$ is obtained from the solution of the time-independent Schr\"odinger equation. A graph must have finely tuned tight-binding parameters in order to have ${\cal T}(E)$$=$$1$. This paper addresses classes of weighted graphs which can be tuned, by adjusting a small fraction of the total weights, to be a quantum dragon. We prove that with proper tuning any nanodevice can be a quantum dragon. Three prescriptions are presented to tune a weighted graph into a quantum dragon nanodevice. The implications of the prescriptions for physical nanodevices is discussed.