  In a graph A, for each two arbitrary vertices g, h with
d(g,h)=2,|MAg2h|=mAg2h is introduced the number of edges of A that are closer
to g than to h. We say A is a 2-edge distance-balanced graph if we have
mAg2h=mAh2g. In this article, we verify the concept of these graphs and present
a method to recognize k-edge distance-balanced graphs for k = 2,3 using
existence of either even or odd cycles. Moreover, we investigate situations
under which the Cartesian and lexicographic products lead to 2-edge distance
-balanced graphs. In some subdivision-related graphs 2-edge distance-balanced
property is verified.
