ON THE TOTAL DISTANCE AND DIAMETER OF GRAPHS

Abstract

The total distance (or Wiener index) of a connected graph$G$is the sum of all distances between unordered pairs of vertices of$G$. DeLaViña and Waller [‘Spanning trees with many leaves and average distance’,Electron. J. Combin.15(1) (2008), R33, 14 pp.] conjectured in 2008 that if$G$has diameter$D>2$and order$2D+1$, then the total distance of$G$is at most the total distance of the cycle of the same order. In this note, we prove that this conjecture is true for 2-connected graphs.