I found this on Abgar V:
https://en.wikipedia.org/wiki/Abgar_V#The_historicity_of_the_Abgar_legend
The historicity of the Abgar legend
The Abgar legend has played an important part in the self-definition of
several Eastern churches, but its historicity is extremely doubtful. Two recent
histories of the Church of the East, Baum and Winkler's The Church of the East
and David Wilmshurst's The Martyred Church, have addressed this issue and have
discussed the growth and development of the legend.[12] Alexander Mirkovic also
argued against the historicity of the legend, pointing out at that the legend is
not the only one of its genre. There were many conversion stories coming out of
the Middle East in the 3rd and 4th centuries. In many ways these stories repre-
sent a model for the conversion of Constantine.[13] Notice the similarity be-
tween the Book of Abgar and the conversion of Helena of Adiabene and her son
Ezad II in Josefus Flavius: Jewish Antiquities, XX 2. In the story told by
Josephus there is a Jewish merchant by the name Ananias and the question is of
circumcision of Ezad.
In the Book of Abgar, Ananias is the messenger sent by King Abgar to Jesus.
Ezad’s son was Abgar VII of Edessa (Ostroene). The origin of the story may be
that Ezad, the father of Abgar VII, had exchanged letters with somebody in Jeru-
salem, but more probably with the Nasi Gamaliel than with Jesus.
Author of the idea Ralph Ellis and Tom Verenna back and forth on it at the
next link:
http://tomverenna.wordpress.com/2013/03/04/a-new-theory-that-jesus-was-king-of-edessa-not-so-fast-mr-ellis/