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ON THE TOTAL DISTANCE AND DIAMETER OF GRAPHS

Part of: Graph theory

Published online by Cambridge University Press:  03 May 2018

HONGBO HUA Open the ORCID record for HONGBO HUA [Opens in a new window]
HONGBO HUA*
Affiliation:
Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, PR China email hongbo_hua@163.com, hongbo.hua@gmail.com
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Abstract

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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.

Keywords

total distancediameter2-connected graphs

MSC classification

Primary: 05C12: Distance in graphs
Secondary: 05C35: Extremal problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 1 , August 2018 , pp. 14 - 17
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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