Chapter

Graph Theory in Paris

Part of the series Trends in Mathematics pp 21-36

Automated Results and Conjectures on Average Distance in Graphs

  • Mustapha AouchicheAffiliated withGERAD and Département de mathématiques et de génie industriel, École Polytechnique de Montréal
  • , Pierre HansenAffiliated withGERAD and Service de l’enseignement des méthodes quantitatives de gestion, HEC Montréal

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Abstract

Using the AutoGraphiX 2 system, a systematic study is made on generation and proof of relations of the form
$$ \underline b _n \leqslant \bar l \oplus i \leqslant \bar b_n $$

where \( \bar l \) denotes the average distance between distinct vertices of a connected graph G, i one of the invariants: diameter, radius, girth, maximum, average and minimum degree, \( \underline b _n \) and \( \bar b_n \) are best possible lower and upper bounds, functions of the order n of G and ⊕ ∈ −, + ×, /. In 24 out of 48 cases simple bounds are obtained and proved by the system. In 21 more cases, the system provides bounds, 16 of which are proved by hand.

Keywords

Graph Invariant AGX Conjecture Average Distance