# On the minimum value of sum-Balaban index

## TL;DR

This paper investigates the extremal values of the sum-Balaban index in graphs, establishing bounds, identifying extremal graphs for small sizes, and analyzing asymptotic behavior in specific graph classes.

## Contribution

It determines the asymptotic upper bound for the minimum sum-Balaban index and characterizes extremal graphs, especially within balanced dumbbell graphs.

## Key findings

- Upper bound for minimum sum-Balaban index is approximately 4.47934 as n→∞.
- Extremal graphs for small n resemble dumbbell graphs with slight modifications.
- Balanced dumbbell graphs with specific clique sizes asymptotically minimize the sum-Balaban index.

## Abstract

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.