Distance-unbalancedness of graphs

•In this manuscript we initiated the study of new structural invariant for graphs, based on a previously studied notion of being (ℓ)-distance-balanced.•We believe that this invariant will attract attention of numerous researchers working with various graph indices, such as the Wiener index, the Szeged index, etc. In this paper we propose and study a new structural invariant for graphs, called distance-unbalancedness, as a measure of how much a graph is (un)balanced in terms of distances. Explicit formulas are presented for several classes of well-known graphs. Distance-unbalancedness of trees is also studied. A few conjectures are stated and some open problems are proposed.