Spectral radius of bipartite graphs

Liu, Chia An; Weng, Chih-wen

doi:10.1016/j.laa.2015.01.040

Cited by 20 publications

(9 citation statements)

References 14 publications

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“…Additionally, we require two propositions, one regarding the largest spectral radius of subgraphs of K p,q of a given size, and another regarding the largest gap between sizes which correspond to a complete bipartite graph of order at most n. Let K m p,q , 0 ≤ pq − m < min{p, q}, be the subgraph of K p,q resulting from removing pq − m edges all incident to some vertex in the larger side of the bipartition (if p = q, the vertex can be from either set). In [17], the authors proved the following result.…”

Section: The Bipartite Spread Conjecturementioning

confidence: 95%

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: The Bipartite Spread Conjecturementioning

confidence: 95%

Maximum spread of graphs and bipartite graphs

Breen¹,

Riasanovsky²,

Tait³

et al. 2021

Preprint

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“…The assumption pq − p < e < pq above ensures that every graph in K(p, q, e) including e K p,q and K e p,q is connected. 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].…”

Section: Introductionmentioning

confidence: 93%

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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“…Let H, H ′ be two bipartite graphs with given ordered bipartitions V H = X Y and V H ′ = X ′ Y ′ , where V H V H ′ = φ. The bipartite sum H + H ′ of H and H ′ (with respect to the given ordered bipartitions) is the graph obtained from H and H ′ by adding an edge between x and y for each pair (x, y) ∈ X × Y ′ X ′ × Y. Chia-an Liu and the third author [10] found upper bounds of ρ(G) expressed by degree sequences of two parts of the bipartition of G.…”

Section: Preliminariesmentioning

confidence: 99%

“…There are several extending results of the above result, which aim to solve an analog of the Brualdi-Hoffman conjecture for nonbipartite graphs [3], proposed in [1]. These extending results are scattered in [1,4,10]. To provide another extending result, we need some notations.…”

Section: Introductionmentioning

confidence: 99%

An Extending Result on Spectral Radius of Bipartite Graphs

Cheng¹,

Fan²,

Weng³

2018

Taiwanese J. Math.

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Let G denote a bipartite graph with e edges without isolated vertices. It was known that the spectral radius of G is at most the square root of e, and the upper bound is attained if and only if G is a complete bipartite graph. Suppose that G is not a complete bipartite graph, and e − 1 and e + 1 are not twin primes. We determine the maximal spectral radius of G. As a byproduct of our study, we obtain a spectral characterization of a pair (e − 1, e + 1) of integers to be a pair of twin primes.

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Spectral radius of bipartite graphs

Cited by 20 publications

References 14 publications

Maximum spread of graphs and bipartite graphs

Maximum spread of graphs and bipartite graphs

Counterexamples of the Bhattacharya-Friedland-Peled conjecture

An Extending Result on Spectral Radius of Bipartite Graphs

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