A Sharp Upper Bound of the Spectral Radius of Graphs

Yan, Hong; Shu, Jinlong; Kun-fu, Fang

doi:10.1006/jctb.2000.1997

Cited by 134 publications

(61 citation statements)

References 5 publications

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“…So, every ℓ-node subgraph in G has at most e ≤ δℓ(ℓ − 1)/2 edges. We now use the bound on the spectral radius of a graph with e edges from [8]: A 2 ≤ √ 2e − n + 1, and since e ≤ δℓ(ℓ − 1)/2, we have that…”

Section: Theorem 2 (No Fptas)mentioning

confidence: 99%

NP-hardness and inapproximability of sparse PCA

Magdon‐Ismail¹

2017

Information Processing Letters

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Section: Theorem 2 (No Fptas)mentioning

confidence: 99%

NP-hardness and inapproximability of sparse PCA

Magdon‐Ismail¹

2017

Information Processing Letters

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“…Using a result of Hong (1988), λ 1 ≤ √ m − n + 1, it is shown in Aouchiche et al (2008a) and Bell (1992) that…”

Section: Conjecture 21mentioning

confidence: 99%

“…Cvetković et al 1995) and the fact that (G) = n −1−δ(G). The second point was proved by using a combination of the following inequalities proved by Stanley (1987) and Hong et al (2001), respectively:…”

Section: This Bound Is Sharp and Attained If And Only If G Or G Is A mentioning

confidence: 99%

Open problems on graph eigenvalues studied with AutoGraphiX

Aouchiche

Caporossi

Hansen

2013

EURO Journal on Computational Optimization

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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.

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“…A bound in terms of the numbers of nodes and links only is found by Hong [9]: if G is connected, then…”

Section: Upper Bounds For the Spectral Radiusmentioning

confidence: 99%

“…Hong, Shu, and Fang [10] obtained a bound that indicates the relation of the spectral radius to the minimal degree: if G is connected, then…”

Section: Upper Bounds For the Spectral Radiusmentioning

confidence: 99%

Robustness of networks against viruses: the role of the spectral radius

Jamakovic

Kooij

Mieghem

et al. 2006

2006 Symposium on Communications and Vehicular Technology

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Abstract-In this paper we study the spectral radius of a number of real-life networks. This study is motivated by the fact that the smaller the spectral radius, the higher the robustness of a network against the spread of viruses. First we study how wellknown upper bounds for the spectral radius of graphs match to the spectral radii of the social network of the Dutch soccer team, the Dutch roadmap network, the network of the observable part of the Internet graph at the IP-level and the Autonomous System level. Secondly, we compare the spectral radius for these real-life networks with those of commonly used complex network models.

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A Sharp Upper Bound of the Spectral Radius of Graphs

Cited by 134 publications

References 5 publications

NP-hardness and inapproximability of sparse PCA

NP-hardness and inapproximability of sparse PCA

Open problems on graph eigenvalues studied with AutoGraphiX

Robustness of networks against viruses: the role of the spectral radius

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