Global efficiency of graphs (Q896087)

The efficiency between two vertices \(i\) and \(j\) is defined to be \(\epsilon_{i,j}=\frac{1}{d(i,j)}\) for all \(i\neq j\). The global efficiency of a graph is defined as \(\displaystyle{E_{\mathrm{glob}}(G)=\frac{1}{n(n-1)}\sum_{i\neq j}\epsilon(v_i,v_j)}\). The authors in this article determine the global efficiency for complete multipartite graphs \(K_{m,n}\), star and subdivided star graphs and the Cartesian products \(K_n\times P_n^m\), \(K_n\times C_n^m\), \(K_n\times K_n\) and \(P_m\times P_n\).