## Abstract

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.

Original language | English |
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Pages (from-to) | 661-676 |

Number of pages | 16 |

Journal | European Journal of Operational Research |

Volume | 191 |

Issue number | 3 |

DOIs | |

Publication status | Published - Dec 16 2008 |

Externally published | Yes |

## Keywords

- Adjacency matrix
- AutoGraphiX
- Conjectures
- Extremal graph
- Graph
- Index
- Irregularity
- Largest eigenvalue
- Spectral spread
- Variable neighborhood search

## ASJC Scopus subject areas

- Information Systems and Management
- General Computer Science
- Industrial and Manufacturing Engineering
- Modelling and Simulation
- Management Science and Operations Research