An Extending Result on Spectral Radius of Bipartite Graphs

Cheng, Yen Jen; Fan, Feng; Weng, Chih-wen

doi:10.11650/tjm/8145

Cited by 7 publications

(8 citation statements)

References 15 publications

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“…Here, 1 is a column all-ones vector, 1

is its transpose (The transposed matrix is the flipped variant of the original matrix. ), and

(·) is the spectral radius of the analyzed vector/matrix defined, as follows [ 30 , 50 ]:

…”

Section: Problem Formulation: Average Consensus Over Mobile Wirelementioning

confidence: 99%

Average Consensus over Mobile Wireless Sensor Networks: Weight Matrix Guaranteeing Convergence without Reconfiguration of Edge Weights

Kenyeres

Kenyeres²

2020

Sensors

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Efficient data aggregation is crucial for mobile wireless sensor networks, as their resources are significantly constrained. Over recent years, the average consensus algorithm has found a wide application in this technology. In this paper, we present a weight matrix simplifying the average consensus algorithm over mobile wireless sensor networks, thereby prolonging the network lifetime as well as ensuring the proper operation of the algorithm. Our contribution results from the theorem stating how the Laplacian spectrum of an undirected simple finite graph changes in the case of adding an arbitrary edge into this graph. We identify that the mixing parameter of Best Constant weights of a complete finite graph with an arbitrary order ensures the convergence in time-varying topologies without any reconfiguration of the edge weights. The presented theorems and lemmas are verified over evolving graphs with various parameters, whereby it is demonstrated that our approach ensures the convergence of the average consensus algorithm over mobile wireless sensor networks in spite of no edge reconfiguration.

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“…Here, 1 is a column all-ones vector, 1

is its transpose (The transposed matrix is the flipped variant of the original matrix. ), and

(·) is the spectral radius of the analyzed vector/matrix defined, as follows [ 30 , 50 ]:

…”

Section: Problem Formulation: Average Consensus Over Mobile Wirelementioning

confidence: 99%

Average Consensus over Mobile Wireless Sensor Networks: Weight Matrix Guaranteeing Convergence without Reconfiguration of Edge Weights

Kenyeres

Kenyeres²

2020

Sensors

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“…It is not difficult to compute the spectrum of each bipartite graph K − p,q or K + p,q [8,20,9]. Proposition 2.6.…”

Section: Corollary 25 For Any Positive Integers P ≤ Q K Pq Is Ds mentioning

confidence: 99%

“…It is direct that for any positive integer p the regular complete bipartite graph K p,p is DS but, for example, the non-isomorphic bipartite graphs K 1,6 and K 2,3 ∪ 2K 1 are cospectral. There are several extending results [1,8,20,9] of the above bound, which aim to solve an analog of the Brualdi-Hoffman conjecture for non-bipartite graphs [3], proposed in [1].…”

Section: Introductionmentioning

confidence: 99%

“…Our research is motivated from the following twin primes bound proposed in [9,Theorem 5.2]: For e ≥ 4, (e − 1, e + 1) is a pair of twin primes if and only if ρ(e) < e + e 2 − 4(e − 1 − √ e − 1) 2 where ρ(e) denotes the maximal ρ(G) of a bipartite graph G on e edges which is not a union of a complete bipartite graph and some isolated vertices. We need to introduce the notations K − p,q and K + p,q of the graphs which are one-edge different from a complete bipartite graph.…”

Section: Introductionmentioning

confidence: 99%

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Spectral characterizations of two families of nearly complete bipartite graphs

Liu

Weng

2017

Annals of Mathematical Sciences and Applications

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It is not hard to find many complete bipartite graphs which are not determined by their spectra. We show that the graph obtained by deleting an edge from a complete bipartite graph is determined by its spectrum. We provide some graphs, each of which is obtained from a complete bipartite graph by adding a vertex and an edge incident on the new vertex and an original vertex, which are not determined by their spectra.

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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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An Extending Result on Spectral Radius of Bipartite Graphs

Cited by 7 publications

References 15 publications

Average Consensus over Mobile Wireless Sensor Networks: Weight Matrix Guaranteeing Convergence without Reconfiguration of Edge Weights

Average Consensus over Mobile Wireless Sensor Networks: Weight Matrix Guaranteeing Convergence without Reconfiguration of Edge Weights

Spectral characterizations of two families of nearly complete bipartite graphs

Counterexamples of the Bhattacharya-Friedland-Peled conjecture

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