Proof of a conjecture on communicability distance sum index of graphs

Let G be a connected graph with adjacency matrix A, and let A=exp⁡(A). The communicability distance between two vertices u and v of G is defined as ξuv=(Auu+Avv−2Auv)1/2, and the communicability distance sum index (CDS index for short) of G is the sum of all communicability distances between vertices of G. In this paper, it is shown that the complete graph Kn is the unique graph attaining the minimum CDS index among all connected graphs of order n. This confirms a conjecture of Estrada (2012) [2]. Furthermore, some upper and lower bounds for the CDS index of graphs are provided.