A connected and nonempty graph A is defined as generalized t-edge distance-balanced, while for each edge f=α\{beta} the number of edges nearer to α than \{beta} are equal to t-times of edges nearer to \{beta} than to α, 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.