  A connected and nonempty graph A is defined as generalized t-edge
distance-balanced, while for each edge f={\alpha}\{beta} the number of edges
nearer to {\alpha} than \{beta} are equal to t-times of edges nearer to \{beta}
than to {\alpha}, for t \in N, or vice versa. We determine some classes of such
graphs. Moreover, we investigate edge-szeged index of the generalized t-edge
distance-balanced graphs. Also, it is discussed about their cartesian and
lexicographic products.
