Proof of a conjecture of V. Nikiforov

Terpai, Tamás

doi:10.1007/s00493-011-2652-1

Cited by 18 publications

(15 citation statements)

References 3 publications

(5 reference statements)

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“…We begin Section 4.1 by mirroring the argument in [30] which proved a conjecture of Nikiforov regarding the largest eigenvalue of a graph and its complement, µ + µ. There Terpai showed that performing two operations on graphons leads to a strict increase in µ + µ.…”

Section: From Graphons To Stepgraphonsmentioning

confidence: 99%

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Maximum spread of graphs and bipartite graphs

Breen¹,

Riasanovsky²,

Tait³

et al. 2021

Preprint

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Given any graph G, the (adjacency) spread of G is the maximum absolute difference between any two eigenvalues of the adjacency matrix of G. In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs. The first states that for all positive integers n, the n-vertex graph G that maximizes spread is the join of a clique and an independent set, with 2n/3 and n/3 vertices, respectively. Using techniques from the theory of graph limits and numerical analysis, we prove this claim for all n sufficiently large. As an intermediate step, we prove an analogous result for a family of operators in the Hilbert space over L 2 [0, 1]. The second conjecture claims that for any fixed e ≤ n 2 /4, if G maximizes spread over all n-vertex graphs with e edges, then G is bipartite. We prove an asymptotic version of this conjecture. Furthermore, we exhibit an infinite family of counterexamples, which shows that our asymptotic solution is tight up to lower order error terms.

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Section: From Graphons To Stepgraphonsmentioning

confidence: 99%

“…Furthermore based on previous work of Nikiforov from [24], the conjecture for graphs reduced directly to maximizing µ + µ for graphons. Using these operations, Terpai [30] reduced to a 4 × 4 stepgraphon and then completed the proof by hand.…”

Section: From Graphons To Stepgraphonsmentioning

confidence: 99%

Maximum spread of graphs and bipartite graphs

Breen¹,

Riasanovsky²,

Tait³

et al. 2021

Preprint

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show abstract

“…Terpai [22] used mathematical analysis and graphons to very nearly solve this conjecture by proving that:…”

Section: Bounds For √ S +mentioning

confidence: 99%

Nordhaus–Gaddum and other bounds for the sum of squares of the positive eigenvalues of a graph

Elphick¹,

Aouchiche²

2017

Linear Algebra and its Applications

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Terpai [22] proved the Nordhaus-Gaddum bound that µ(G) + µ(G) ≤ 4n/3 − 1, where µ(G) is the spectral radius of a graph G with n vertices. Let s + denote the sum of the squares of the positive eigenvalues of G. We prove that s + (G)+ s + (G) < √ 2nand conjecture that s + (G) + s + (G) ≤ 4n/3 − 1. We have used AutoGraphiX and Wolfram Mathematica to search for a counter-example. We also consider Nordhaus-Gaddum bounds for s + and bounds for the Randić index.

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“…The lower bound in (1) is best possible and is attained if and only if G is a regular graph; however the upper bound can be improved significantly. A minor improvement has been shown in [8], but an essentially best possible bound has been found only recently, by Csikvari [4] and Terpai [11] who showed that µ (G) + µ G ≤ 4n/3 − 1.…”

Section: Introductionmentioning

confidence: 97%

Proof of a conjecture of V. Nikiforov

Cited by 18 publications

References 3 publications

Maximum spread of graphs and bipartite graphs

Maximum spread of graphs and bipartite graphs

Nordhaus–Gaddum and other bounds for the sum of squares of the positive eigenvalues of a graph

More eigenvalue problems of Nordhaus–Gaddum type

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