Open problems on graph eigenvalues studied with AutoGraphiX

Since the late forties of the last century, methods of operations research have been extensively used to solve problems in graph theory, and graph theory has been extensively used to model operations research problems and to solve optimization problems on graphs, e.g., shortest paths and network flo...

Descripción completa

Guardado en:

Detalles Bibliográficos
Autores principales:	Mustapha Aouchiche, Gilles Caporossi, Pierre Hansen
Formato:	article
Lenguaje:	EN
Publicado:	Elsevier 2013
Materias:	Primary 05C50 Applied mathematics. Quantitative methods T57-57.97 Electronic computers. Computer science QA75.5-76.95
Acceso en línea:	https://doaj.org/article/4142b823c511428fb6c3742809eacd9d
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

id	oai:doaj.org-article:4142b823c511428fb6c3742809eacd9d
record_format	dspace
spelling	oai:doaj.org-article:4142b823c511428fb6c3742809eacd9d2021-12-02T05:00:36ZOpen problems on graph eigenvalues studied with AutoGraphiX2192-440610.1007/s13675-012-0001-9https://doaj.org/article/4142b823c511428fb6c3742809eacd9d2013-05-01T00:00:00Zhttp://www.sciencedirect.com/science/article/pii/S2192440621000174https://doaj.org/toc/2192-4406Since the late forties of the last century, methods of operations research have been extensively used to solve problems in graph theory, and graph theory has been extensively used to model operations research problems and to solve optimization problems on graphs, e.g., shortest paths and network flow problems. More recently, methods of operations research and artificial intelligence have been used to advance graph theory per se, i.e., to find conjectures on graph theory invariants, to refute such conjectures and in some cases to find automated proofs or ideas of proofs. Among other systems, the AutoGraphiX system was developed since 1997 at GERAD (Montreal) by the present authors. Extensive experiments have been conducted which led to 1,700 conjectures, about 800 of which turned out to be easy and could be proved by the system, and about 600 further ones were proved by hand by us or graph theorists from various countries. Moreover, these results led to many generalizations and further papers. In this paper, we study four theoretical problems related to the eigenvalues of (the adjacency matrix of) a connected graph and to which AutoGraphiX was applied. Three of the problems are related to the maximum value of the irregularity, the maximum spectral spread and the upper bound of Nordhaus–Gaddum type on the index, over the class of connected graphs on n vertices. The fourth problem concerns the maximization of the energy (the sum of the absolute values of the eigenvalues) of a connected graph with fixed numbers of vertices and of cycles. We present a brief survey of the papers on or in connection with these problems, and give some new partial results.Mustapha AouchicheGilles CaporossiPierre HansenElsevierarticlePrimary 05C50Applied mathematics. Quantitative methodsT57-57.97Electronic computers. Computer scienceQA75.5-76.95ENEURO Journal on Computational Optimization, Vol 1, Iss 1, Pp 181-199 (2013)
institution	DOAJ
collection	DOAJ
language	EN
topic	Primary 05C50 Applied mathematics. Quantitative methods T57-57.97 Electronic computers. Computer science QA75.5-76.95
spellingShingle	Primary 05C50 Applied mathematics. Quantitative methods T57-57.97 Electronic computers. Computer science QA75.5-76.95 Mustapha Aouchiche Gilles Caporossi Pierre Hansen Open problems on graph eigenvalues studied with AutoGraphiX
description	Since the late forties of the last century, methods of operations research have been extensively used to solve problems in graph theory, and graph theory has been extensively used to model operations research problems and to solve optimization problems on graphs, e.g., shortest paths and network flow problems. More recently, methods of operations research and artificial intelligence have been used to advance graph theory per se, i.e., to find conjectures on graph theory invariants, to refute such conjectures and in some cases to find automated proofs or ideas of proofs. Among other systems, the AutoGraphiX system was developed since 1997 at GERAD (Montreal) by the present authors. Extensive experiments have been conducted which led to 1,700 conjectures, about 800 of which turned out to be easy and could be proved by the system, and about 600 further ones were proved by hand by us or graph theorists from various countries. Moreover, these results led to many generalizations and further papers. In this paper, we study four theoretical problems related to the eigenvalues of (the adjacency matrix of) a connected graph and to which AutoGraphiX was applied. Three of the problems are related to the maximum value of the irregularity, the maximum spectral spread and the upper bound of Nordhaus–Gaddum type on the index, over the class of connected graphs on n vertices. The fourth problem concerns the maximization of the energy (the sum of the absolute values of the eigenvalues) of a connected graph with fixed numbers of vertices and of cycles. We present a brief survey of the papers on or in connection with these problems, and give some new partial results.
format	article
author	Mustapha Aouchiche Gilles Caporossi Pierre Hansen
author_facet	Mustapha Aouchiche Gilles Caporossi Pierre Hansen
author_sort	Mustapha Aouchiche
title	Open problems on graph eigenvalues studied with AutoGraphiX
title_short	Open problems on graph eigenvalues studied with AutoGraphiX
title_full	Open problems on graph eigenvalues studied with AutoGraphiX
title_fullStr	Open problems on graph eigenvalues studied with AutoGraphiX
title_full_unstemmed	Open problems on graph eigenvalues studied with AutoGraphiX
title_sort	open problems on graph eigenvalues studied with autographix
publisher	Elsevier
publishDate	2013
url	https://doaj.org/article/4142b823c511428fb6c3742809eacd9d
work_keys_str_mv	AT mustaphaaouchiche openproblemsongrapheigenvaluesstudiedwithautographix AT gillescaporossi openproblemsongrapheigenvaluesstudiedwithautographix AT pierrehansen openproblemsongrapheigenvaluesstudiedwithautographix
_version_	1718400823903387648

Open problems on graph eigenvalues studied with AutoGraphiX

Ejemplares similares