Graphs with least eigenvalue \(-2\); a historical survey and recent developments in maximal exceptional graphs
type: Item
instance of: scholarly article
title: Graphs with least eigenvalue \(-2\); a historical survey and recent developments in maximal exceptional graphs
zbMATH Open document ID: 1012.05113
DOI: 10.1016/S0024-3795(02)00377-4
published in: Linear Algebra and its Applications
publication date: 2003-02-05T00:00:00Z
review text: Connected graphs with least eigenvalue at least \(-2\) are either generalized line graphs or so-called exceptional graphs. The author surveys a variety of results in the study of these graphs and reports on the recent characterization of all (maximal) exceptional graphs independently accomplished by two groups of researchers one of which containing him. The basic techniques used for this characterization are the use of representations of the graphs using root systems known from the theory of Lie algebras and the so-called star complement technique. The author sketches a proof that all exceptional graphs, except for the cone over \(L(K_8)\), can be obtained by the star complement technique from a unique (exceptional) star complement for the eigenvalue \(-2\).
reviewed by: Dieter Rautenbach
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C75
zbMATH DE Number: 1864761
zbMATH Keywords: eigenvalue
zbMATH Keywords: spectrum of a graph
zbMATH Keywords: generalized line graph
zbMATH Keywords: exceptional graph
zbMATH Keywords: root systems
zbMATH Keywords: star complement technique
zbMATH Keywords: characterization
author: Dragoš Cvetković
MaRDI profile type: Publication
Wikidata QID: Q54066090
cites work: Characterizations of derived graphs
cites work: Multiplicative cones - a family of three eigenvalue graphs
cites work: Q3992965
cites work: Q4105702
cites work: Q4173399
cites work: Exceptional graphs with smallest eigenvalue -2 and related problems
cites work: A note on generalized line graphs
cites work: Line graphs, root systems, and elliptic geometry
cites work: Q4000288
cites work: Q5662096
cites work: Q3262641
cites work: Q3270191
cites work: The Uniqueness of the Triangular Association Scheme
cites work: Q5648382
cites work: Q3924243
cites work: Q3315545
cites work: Recent results in the theory of graph spectra
cites work: Q4327350
cites work: Q3872509
cites work: Generalized line graphs
cites work: Q4796686
cites work: The maximal exceptional graphs
cites work: A table of connected graphs on six vertices
cites work: Q3312270
cites work: Eigenspaces of graphs
cites work: Some characterizations of graphs by star complements
cites work: Graphs with least eigenvalue \(-2\): The star complement technique
cites work: Q3153189
cites work: Nonregular graphs with three eigenvalues
cites work: On characterizing certain graphs with four eigenvalues by their spectra
cites work: On spectral characterization and embeddings of graphs
cites work: Q4286076
cites work: On the Exceptional Case in a Characterization of the Arcs of a Complete Graph
cites work: On the Uniqueness of the Triangular Association Scheme
cites work: On the Line Graph of a Projective Plane
cites work: Q5553120
cites work: Graphs with eigenvalues at least - 2
cites work: Q4043127
cites work: Q3956990
cites work: Q4281313
cites work: On graphs with multiple eigenvalues
cites work: Q4511043
cites work: Strongly regular graphs with (-1, 1, 0) adjacency matrix having eigenvalue 3
cites work: Q5181734
cites work: Q4065580
cites work: On a Characterization of the Triangular Association Scheme
cites work: Q3765811
full work available at URL: s0024-3795(02)00377-4
OpenAlex ID: W2077586574
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