A graph G is \(\ell \)-distance-balanced if for each pair of vertices x and y at a distance \(\ell \) in G, the number of vertices closer to x than to y is equal to the number of vertices closer to y than to x. A complete characterization of \(\ell \)-distance-balanced corona products is given and characterization of lexicographic products for \(\ell \ge 3\), thus complementing known results for \(\ell \in \{1,2\}\) and correcting an earlier related assertion. A sufficient condition on H which guarantees that \(K_n\,\square \,H\) is \(\ell \)-distance-balanced is given, and it is proved that if \(K_n\,\square \,H\) is \(\ell \)-distance-balanced, then H is an \(\ell \)-distance-balanced graph. A known characterization of 1-distance-balanced graphs is extended to \(\ell \)-distance-balanced graphs, again correcting an earlier claimed assertion.

一个图ģ是\（\ ELL \） -distance平衡如果对于每对顶点X和ÿ在距离\（\ ELL \）在ģ，顶点的数目更接近X比ý等于数接近y而不是接近x的顶点数量。给出了\（\ ell \） -距离平衡的电晕产物的完整表征，并给出了\（\ ell \ ge 3 \）的词典编合物的表征，从而补充了\（\ ell \ in \ {1,2 \} \）并更正先前的相关断言。H的充分条件保证\（K_n \，\ square \，H \）是\（\ ell \） -距离平衡，并证明如果\（K_n \，\ square \，H \）是\（ \ ell \） -距离平衡图，则H是\（\ ell \） -距离平衡图。1-距离平衡图的已知特征扩展到\（\ ell \） -距离平衡图，再次校正了先前要求保护的断言。