Discussiones Mathematicae Graph Theory 29(1) (2009) 15-37
doi: 10.7151/dmgt.1430

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VARIABLE NEIGHBORHOOD SEARCH FOR EXTREMAL GRAPHS. 17. FURTHER CONJECTURES AND RESULTS ABOUT THE INDEX

Mustapha Aouchiche

HEC Montréal
3000 Chemin de la Cote-Sainte-Catherine
Montréal, Canada
e-mail: mustapha.aouchiche@gerad.ca

Pierre Hansen

GERAD and HEC Montréal
3000 Chemin de la Cote-Sainte-Catherine
Montréal, Canada
e-mail: pierre.hansen@gerad.ca

Dragan Stevanović

PINT, University of Primorska, Slovenia
e-mail: dragance106@yahoo.com

Abstract

The AutoGraphiX 2 system is used to compare the index of a connected graph G with a number of other graph theoretical invariants, i.e., chromatic number, maximum, minimum and average degree, diameter, radius, average distance, independence and domination numbers. In each case, best possible lower and upper bounds, in terms of the order of G, are sought for sums, differences, ratios and products of the index and another invariant. There are 72 cases altogether: in 7 cases known results were reproduced, in 32 cases immediate results were obtained and automatically proved by the system, conjectures were obtained in 27 cases, of which 12 were proved (in 3 theorems and 9 propositions), 9 remain open and 6 were refuted. No results could be derived in 7 cases.

Keywords: AutoGraphiX, automated conjecture making, index of a graph, spectral radius, graph invariant.

2000 Mathematics Subject Classification: 05C35, 05C50, 05C75.

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Received 23 July 2007
Revised 16 October 2008
Accepted 16 October 2008