  We consider extremal values of sum-Balaban index among graphs on $n$
vertices. We determine that the upper bound for the minimum value of the
sum-Balaban index is at most $4.47934$ when $n$ goes to infinity. For small
values of $n$ we determine the extremal graphs and we observe that they are
similar to dumbbell graphs, in most cases having one extra edge added to the
corresponding extreme for the usual Balaban index. We show that in the class of
balanced dumbbell graphs, those with clique sizes $\sqrt[4]{\sqrt
2\log\big(1+\sqrt 2\big)}\sqrt n+o(\sqrt n)$ have asymptotically the smallest
value of sum-Balaban index. We pose several conjectures and problems regarding
this topic.
