Bounds on the radius and status of graphs

Two classical concepts of centrality in a graph are the median and the center. The connected notions of the status and the radius of a graph seem to be in no relation. In this paper, however, we show a clear connection of both concepts, as they obtain their minimum and maximum values at the same type of tree graphs. Trees with fixed maximum degree and extremum radius and status, resp., are characterized. The bounds on radius and status can be transferred to general connected graphs via spanning trees. A new method of proof allows not only to regain results of Lin et al. on graphs with extremum status, but it allows also to prove analogous results on graphs with extremum radius

Neighbor Joining And Leaf Status

The Neighbor Joining Algorithm is among the most fundamental algorithmic results in computational biology. However, its definition and correctness proof are not straightforward. In particular, ''the question ''what does the NJ method seek to do?'' has until recently proved somewhat elusive'' [Gascuel \& Steel, 2006]. While a rigorous mathematical analysis is now available, it is still considered somewhat hard to follow and its proof tedious at best. In this work, we present an alternative interpretation of the goal of the Neighbor Joining algorithm by proving that it chooses to merge the two taxa u and v that maximize the ''leaf-status'', that is, the sum of distances of all leaves to the unique u-v-path