The Wiener, eccentric connectivity and Zagreb indices of the hierarchical product of graphs (Q2871020)

Let \(G_1\) and \(G_2\) be two graphs with vertex sets \(V_1\) and \(V_2\) having a distinguished or root vertex, labeled 0. The hierarchical product of them is defined as the graph with vertices the tuples \(x_2x_1\), with \(x_i\in V_i\) (\(i=1,2\)). Edges are defined as follows: \(x_2x_1\sim x_2y_1\) if \(y_1x_1\) is an edge in \(G_1\); \(x_2x_1\sim y_2x_1\) if \(y_2x_2\) is an edge in \(G_2\) and \(x_1=0\). Some graph invariants such as the Zagreb index, Wiener index, and eccentric connectivity are computed for this graph operation.