Mustapha Aouchiche

A survey of Nordhaus-Gaddum type relations

In 1956, Nordhaus and Gaddum gave lower and upper bounds on the sum and the product of the chromatic number of a graph and its complement, in terms of the order of the graph. Since then, relations of a similar type have been proposed for many other graph invariants, in several hundred papers. We present a survey on this research endeavor.

Variable Neighborhood Search for Extremal Graphs 14: The AutoGraphiX 2 System

The AutoGraphiX (AGX) system for computer assisted or, for some of its functions, fully automated graph theory was developed at GERAD, Montreal since 1997. We report here on a new version (AGX 2) of that system. It contains many enhancements, as well as a new function for automated proof of simple propositions. Among other results, AGX 2 led to several hundred new conjectures, ranking from easy ones, proved automatically, to others requiring longer unassisted or partially assisted proofs, to open ones. Many examples are given, illustrating AGX 2’s functions and the results obtained.

On a conjecture about the Szeged index

Khalifeh, Yousefi-Azari, Ashrafi and Wagner [M.K. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, S.G. Wagner, Some new results on distance-based graph invariants, European J. Combin. 30 (2009) 1149-1163] conjectured that for a connected graph G on n vertices and m edges with Szeged index Sz, Sz=mn^2/4 if and only if G is a regular bipartite graph. In this note, we disprove this conjecture and then prove a stronger result from which it follows that the equality holds if and only if G is a transmission-regular bipartite graph.

Variable neighborhood search for extremal graphs. 16. Some conjectures related to the largest eigenvalue of a graph

We consider four conjectures related to the largest eigenvalue of (the adjacency matrix of) a graph (i.e., to the index of the graph). Three of them have been formulated after some experiments with the programming system AutoGraphiX, designed for finding extremal graphs with respect to given properties by the use of variable neighborhood search. The conjectures are related to the maximal value of the irregularity and spectral spread in n-vertex graphs, to a Nordhaus-Gaddum type upper bound for the index, and to the maximal value of the index for graphs with given numbers of vertices and edges. None of the conjectures has been resolved so far. We present partial results and provide some indications that the conjectures are very hard.

Variable neighborhood search for extremal graphs. 17. Further conjectures and results about the index

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: ∗This work has been supported by NSERC Grant 105574–2002 and the Data Mining Chair of HEC Montréal, Canada. The third author acknowledges partial support by Grant 144015G of Serbian Ministry of Science. 16 M. Aouchiche, P. Hansen and D. Stevanović 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.

AutoGraphiX: a survey

Abstract A survey is made of the AutoGraphiX (AGX) research programme for computer assisted and, for some functions, automated graph theory.

Automated Results and Conjectures on Average Distance in Graphs

Using the AutoGraphiX 2 system, a systematic study is made on generation and proof of relations of the form

Proximity and remoteness in graphs: Results and conjectures

\underline b _n \leqslant \bar l \oplus i \leqslant \bar b_n