## Abstract

Balaban index is defined as *G*, *n* and *m* are the cardinalities of the vertex and the edge set of *G*, respectively, and *w*(*u*) (resp. *w*(*v*)) denotes the sum of distances from *u* (resp. *v*) to all the other vertices of *G*. In 2011, H. Deng found an interesting property that Balaban index is a convex function in double stars. We show that this holds surprisingly to general graphs by proving that attaching leaves at two vertices in a graph yields a new convexity property of Balaban index. We demonstrate this property by finding, for each *n*, seven trees with the maximum value of Balaban index, and we conclude the paper with an interesting conjecture.