Nonstandard digraphs and transfinite digraphs have been defined and examined in two prior technical reports. The present work examines digraphs that are both nonstandard and transfinite. This requires a combination in certain ways of the techniques used in the prior two works. We first construct herein nonstandard transfinite digraphs for all the natural-number ranks. Then, a special kind of nonstandard transfinite digraph having the "arrow rank" $\vec{ \omega}$ needs to be constructed. Once this is done, the first limit-ordinal rank $\omega$ can be attained. This procedure can be continued on to still higher ranks.
Categories: math.CO