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ON THE TOTAL DISTANCE AND DIAMETER OF GRAPHS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 98 / Issue 1 / August 2018
- Published online by Cambridge University Press:
- 03 May 2018, pp. 14-17
- Print publication:
- August 2018
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- Article
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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.
