We theoretically and experimentally demonstrate that graded-index (GI) core polymer optical waveguides are a promising component realizing high-speed and high-density on-board interconnects. As a fabrication method of the GI circular core polymer optical waveguides, we introduce the Mosquito method, in which we utilize a micro dispenser. By the Mosquito method, GI circular core polymer parallel optical waveguides with low-loss and low inter-channel crosstalk are fabricated successfully without using any types of photo mask. We also theoretically show that the GI-core polymer waveguides enable remarkably low loss waveguide circuits involving waveguide crossings in a mono layer. Although the Mosquito method is not suitable for the crossed waveguide fabrication, we show an alternative technique to realize the low loss GI-core crossed waveguide: the Photo-addressing method.