The graphs with spectral radius between 2 and 2+5

Brouwer, AE Andries; Neumaier, Arnold

doi:10.1016/0024-3795(89)90466-7

Cited by 57 publications

(122 citation statements)

References 2 publications

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“…The other case (e = 2) is similar. As mentioned before, 2,3 for n large enough. Then by Theorem 3.4, G n is a subgraph of a Laundry graph or an Urchin graph, for n large enough.…”

Section: Van Dam and Kooijsupporting

confidence: 53%

“…The first two are classical results by Smith [10], and the third result is by Brouwer and Neumaier [2]. The results require the following definitions.…”

Section: Lemma 24 Let Uv Be An Edge Of a Connected Graph G If Uv Imentioning

confidence: 94%

“…For n 6(e + 2) such that n − e is even, take the graph H n = C . Since ρ(G n ) ρ(H n ) we can take > 0 such that ρ(G n ) ρ 1 + < θ 2,3 for n large enough. By Theorem 3.4, G n is a subgraph of a Laundry graph or an Urchin graph, for n large enough.…”

Section: Van Dam and Kooijmentioning

confidence: 99%

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Asymptotic Results on the Spectral Radius and the Diameter of Graphs

et al. 2008

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Asymptotic results on the spectral radius and the diameter of graphs Cioaba, S.M.; van Dam, Edwin; Koolen, J.H.; Lee, J.H. Published in: Linear Algebra and its ApplicationsPublication date: 2010 Link to publication General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.-Users may download and print one copy of any publication from the public portal for the purpose of private study or research -You may not further distribute the material or use it for any profit-making activity or commercial gain -You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim. We study the limit points of the spectral radii of certain families of graphs, and apply the results to the problem of minimizing the spectral radius among the graphs with a given number of vertices and diameter. In particular, we consider the cases when the diameter is about half the number of vertices, and when the diameter is near the number of vertices. We prove certain instances of a conjecture posed by Van Dam and Kooij [E.R. Van Dam, R.E. Kooij, The minimal spectral radius of graphs with a given diameter, Linear Algebra Appl. 423 (2007) [408][409][410][411][412][413][414][415][416][417][418][419] and show that the conjecture is false for the other instances. A R T I C L E I N F O A B S T R A C T

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“…The other case (e = 2) is similar. As mentioned before, 2,3 for n large enough. Then by Theorem 3.4, G n is a subgraph of a Laundry graph or an Urchin graph, for n large enough.…”

Section: Van Dam and Kooijsupporting

confidence: 53%

“…The first two are classical results by Smith [10], and the third result is by Brouwer and Neumaier [2]. The results require the following definitions.…”

Section: Lemma 24 Let Uv Be An Edge Of a Connected Graph G If Uv Imentioning

confidence: 94%

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Asymptotic Results on the Spectral Radius and the Diameter of Graphs

et al. 2008

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“…There is also a classification of graphs, similar to Smith's, for graphs having spectral radius in the interval (2, 2 + √ 5] due to Cvetković, Doob, and Gutman [15] and Brouwer and Neumaier [12]. From this, we easily obtain the following.…”

Section: Theorem 43 (Smith)mentioning

confidence: 83%

On groups generated by two positive multi-twists: Teichmueller curves and Lehmer’s number

Leininger

2004

Geom. Topol.

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From a simple observation about a construction of Thurston, we derive several interesting facts about subgroups of the mapping class group generated by two positive multi-twists. In particular, we identify all configurations of curves for which the corresponding groups fail to be free, and show that a subset of these determine the same set of Teichmüller curves as the non-obtuse lattice triangles which were classified by Kenyon, Smillie, and Puchta. We also identify a pseudo-Anosov automorphism whose dilatation is Lehmer's number, and show that this is minimal for the groups under consideration. In addition, we describe a connection to work of McMullen on Coxeter groups and related work of Hironaka on a construction of an interesting class of fibered links.

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“…Subsequently, Cvetković et al [2] gave a nearly complete description of all graphs G with 2 < λ 1 ≤ 2 + √ 5 and their description was completed by Brouwer and Neumaier [1]. Later Woo and Neumaier [16] examine the structure of graphs G with 2 + √ 5 < λ 1 ≤ 3 2 √ 2 and their result is presented in Section 2.…”

mentioning

confidence: 99%