This paper describes the production of graphene nanocomposites via melt mixing of thermally reduced graphite oxide with ethylene vinyl-acetate copolymers of different (0-70 wt%) vinyl acetate content, and their measured electrical and rheological properties. The aim of these studies was to investigate the influence of a continually changing polymer matrix polarity on the dispersion and percolating behavior of graphene fillers, an effect that can be expected to be most prominent with the high specific surfaces of the latter. Composites with graphite and multi-walled carbon nanotubes were produced and examined for comparison. The effectivity of the dispersion process was checked by measuring the melt rheology and electrical conductivity of the samples. The percolation thresholds derived from these measurements show a minimum for VA contents around 20 wt%. The thresholds for electrical conductivity are by a factor around 1.5 lower than the rheological values, and both are distinctively higher than those observed from composites produced via solution mixing. The percolation behavior is compared to predictions made from the surface energy of the compounds. [ABSTRACT FROM AUTHOR]