LAUSR

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