Proof of conjectures on remoteness and proximity in graphs

The remoteness ρ=ρ(G) of a connected graph G is the maximum, over all vertices, of the average distance from a vertex to all others, while the proximity π=π(G) of a connected graph G is the minimum, over all vertices, of the average distance from a vertex to all others. In this paper, we first deal with some conjectures on remoteness and proximity, among which two conjectures were proved, while the other two conjectures were disproved by counter examples. Then we obtain some new upper bounds for remoteness and proximity in terms of some graph invariants. Moreover, we use remoteness to give a new sufficient condition for a connected bipartite graph to be Hamiltonian.