Summary: The harmonic index \(H(G)\) of a graph \(G\) is defined as the sum of the weights \(\frac{2}{d(u)+d(v)}\) of all edges \(uv\) of \(G\), where \(d(u)\) denotes the degree of the vertex \(u\) in \(G\). In this work we compute the harmonic index of a graph with a cut-vertex and with more than one cut-vertex. As an application, this topological index is computed for Bethe trees and dendrimer trees. Also, the harmonic indices of fasciagraph and a special type of trees, namely, polytree are computed.