On the minimum value of sum-Balaban index

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.