In this master's thesis we firstly introduce the concept of a distance-balanced graph. Then we present some examples of distance-balanced graphs and a family of graphs, no member of which is distance-balanced. We then make a natural generalization of this concept to define l-distance-balanced graphs. Here, as an example, we look at Cayley graphs of abelian groups, which are all highly distance-balanced. Then we focus on generalized Petersen graphs. In the second main part of the thesis we generalize the so-called Mostar index to define the distance-unbalancedness index of a graph. Here we first compute this index for paths and complete multipartite graphs and then also for some lesser-known families of graphs. After that we present a few results and conjectures regarding the smallest and the second smallest possible value of the distance-unbalancedness index for trees of given order. Finally, we present the proof of the theorem showing that among all trees of order n, where n>=5, the star S_{n-1} is the unique tree of order n with the smallest possible value of the distance-unbalancedness index.
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