Spectral characterizations of two families of nearly complete bipartite graphs

Liu, Chia-an; Weng, Chih-wen

doi:10.4310/amsa.2017.v2.n2.a2

Cited by 5 publications

(1 citation statement)

References 18 publications

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“…Conjecture 1.2 was proved by Liu and Weng [9] in 2015. For applications, there are extending results on the spectral characterization of the nearly complete bipartite graphs [6,10]. However, as e ≤ pq − p, things have changed.…”

Section: Introductionmentioning

confidence: 99%

Counterexamples of the Bhattacharya-Friedland-Peled conjecture

Cheng¹,

Liu²,

Weng³

2021

Preprint

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The Brauldi-Hoffman conjecture, proved by Rowlinson in 1988, characterized the graph with maximal spectral radius among all simple graphs with prescribed number of edges. In 2008, Bhattacharya, Friedland, and Peled proposed an analog, which will be called the BFP conjecture in the following, of the Brauldi-Hoffman conjecture for the bipartite graphs with fixed numbers of edges in the graph and vertices in the bipartition. The BFP conjecture was proved to be correct if the number of edges is large enough by several authors. However, in this paper we provide some counterexamples of the BFP conjecture.

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Section: Introductionmentioning

confidence: 99%

Counterexamples of the Bhattacharya-Friedland-Peled conjecture

Cheng¹,

Liu²,

Weng³

2021

Preprint

Self Cite

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Counterexamples of the Bhattacharya-Friedland-Peled conjecture

Cheng

Liu

Weng

2022

Linear Algebra and its Applications

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Geometry of error amplification in solving the Prony system with near-colliding nodes

2020

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We consider a reconstruction problem for “spike-train” signals F F of an a priori known form F ( x ) = ∑ j = 1 d a j δ ( x − x j ) , F(x)=\sum _{j=1}^{d}a_{j}\delta \left (x-x_{j}\right ), from their moments m k ( F ) = ∫ x k F ( x ) d x . m_k(F)=\int x^kF(x)dx. We assume that the moments m k ( F ) m_k(F) , k = 0 , 1 , … , 2 d − 1 k=0,1,\ldots ,2d-1 , are known with an absolute error not exceeding ϵ > 0 \epsilon > 0 . This problem is essentially equivalent to solving the Prony system ∑ j = 1 d a j x j k = m k ( F ) , k = 0 , 1 , … , 2 d − 1. \sum _{j=1}^d a_jx_j^k=m_k(F), \ k=0,1,\ldots ,2d-1. We study the “geometry of error amplification” in reconstruction of F F from m k ( F ) , m_k(F), in situations where the nodes x 1 , … , x d x_1,\ldots ,x_d near-collide, i.e., form a cluster of size h ≪ 1 h \ll 1 . We show that in this case, error amplification is governed by certain algebraic varieties in the parameter space of signals F F , which we call the “Prony varieties”. Based on this we produce lower and upper bounds, of the same order, on the worst case reconstruction error. In addition we derive separate lower and upper bounds on the reconstruction of the amplitudes and the nodes. Finally we discuss how to use the geometry of the Prony varieties to improve the reconstruction accuracy given additional a priori information.

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Spectral characterizations of two families of nearly complete bipartite graphs

Cited by 5 publications

References 18 publications

Counterexamples of the Bhattacharya-Friedland-Peled conjecture

Counterexamples of the Bhattacharya-Friedland-Peled conjecture

Counterexamples of the Bhattacharya-Friedland-Peled conjecture

Geometry of error amplification in solving the Prony system with near-colliding nodes

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