Spectral radius of weighted adjacency matrix of trees based on Balaban index

For a graph [Formula: see text], the sum of the distance between vertex [Formula: see text] and other vertices in [Formula: see text] is denoted by [Formula: see text]. The Balaban index of [Formula: see text] is defined as [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text]. Based on the Balaban index of a graph, a weighted function is defined as [Formula: see text], [Formula: see text]. Define a matrix [Formula: see text] as the weighted adjacency matrix of [Formula: see text], where [Formula: see text] if [Formula: see text] and [Formula: see text] otherwise. In this work, we show that star [Formula: see text] is with the largest spectral radius of [Formula: see text] among trees with [Formula: see text] vertices. Inversely, path [Formula: see text] is with the smallest spectral radius of [Formula: see text] in all caterpillar trees with [Formula: see text] vertices.