The communicability distance in graphs

Let G be a simple connected graph with adjacency matrix A. The communicabilityGpq between two nodes p and q of the graph is defined as the pq-entry of G=exp(A). We prove here that ξp,q=(Gpp+Gqq-2Gpq)1/2 is a Euclidean distance and give expressions for it in paths, cycles, stars and complete graphs with n nodes. The sum of all communicabilitydistances in a graph is introduced as a new graph invariant ϒ(G). We compare this index with the Wiener and Kirchhoff indices of graphs and conjecture about the graphs with maximum and minimum values of this index.