Proximity and Remoteness in Graphs: Results and Conjectures

The proximity of a connected graph G is the minimum, over all vertices, of the average distance from a vertex to all others. Similarly, the maximum is called the remoteness and denoted by . In this paper we first prove upper and lower bounds on and as a function of the order n of G. A comparison between these two invariants follows and then each one is compared to the diameter, radius, average eccentricity, average distance, independence number and matching number. Most bounds so obtained are proved, but a few of them remain open conjectures.