Graphs with least eigenvalue −2: Ten years on

Cvetković, Dragoš; Rowlinson, Peter; Simić, Slavko

doi:10.1016/j.laa.2015.06.012

Cited by 16 publications

(7 citation statements)

References 126 publications

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“…To fix ideas, suppose the cycle C = x → y, y → z, z → t, t → z exists in G, and consider the vector v with ). The dimension of the space spanned by the even-length cycles has been studied in [5,6,9], and it is known to be m − n when G is not bipartite and m − n + 1 when it is.…”

Section: Proposition 414 Let G Have At Least Two Cycles Then Ammentioning

confidence: 99%

Non-backtracking Spectrum: Unitary Eigenvalues and Diagonalizability

Torres

2020

Preprint

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Much effort has been spent on characterizing the spectrum of the non-backtracking matrix of certain classes of graphs, with special emphasis on the leading eigenvalue or the second eigenvector. Much less attention has been paid to the eigenvalues of small magnitude; here, we fully characterize the eigenvalues with magnitude equal to one. We relate the multiplicities of such eigenvalues to the existence of specific subgraphs. We formulate a conjecture on necessary and sufficient conditions for the diagonalizability of the non-backtracking matrix. As an application, we establish an interlacing-type result for the Perron eigenvalue.

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Section: Proposition 414 Let G Have At Least Two Cycles Then Ammentioning

confidence: 99%

Non-backtracking Spectrum: Unitary Eigenvalues and Diagonalizability

Torres

2020

Preprint

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“…It is easy to check that ǫd i+1 > n, implying that M ≤ i similarly to the proof of Claim 17. It follows from Inequalities (5), (6) and Claim 13 that…”

Section: The Setsmentioning

confidence: 96%

“…In the last few decades much research has been done on spectra of graphs, i.e. the eigenvalues of the adjacency matrices of graphs; see Finck and Grohmann [10], Hoffman [16,17], Nosal [25], Cvetković, Doob and Sachs [7], Neumaier [20], Brigham and Dutton [3,4], Brualdi and Hoffman [5], Stanley [30], Shearer [29], Powers [26], Favaron, Mahéo and Saclé [8,9], Hong [18], Liu, Shen and Wang [19], Nikiforov [22,23,24,21], and Cvetković, Rowlinson and Simić [6] for a small selection of relevant publications. Perhaps the most basic property of the spectrum of a graph is its radius, i.e.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues of subgraphs of the cube

Bollobás

Letzter

2018

European Journal of Combinatorics

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We consider the problem of maximising the largest eigenvalue of subgraphs of the hypercube Q d of a given order. We believe that in most cases, Hamming balls are maximisers, and our results support this belief. We show that the Hamming balls of radius o(d) have largest eigenvalue that is within 1 + o(1) of the maximum value. We also prove that Hamming balls with fixed radius maximise the largest eigenvalue exactly, rather than asymptotically, when d is sufficiently large. Our proofs rely on the method of compressions.

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“…The second result of Hoffman [48] shows that the smallest eigenvalue of a graph depends very much on its local structure. For recent surveys on graphs with smallest eigenvalue at least −2, we refer to Cvetković, Rowlinson and Simić [33] and [34]. In Section 3, we define Hoffman graphs and present the basic theory of Hoffman graphs.…”

Section: Introductionmentioning

confidence: 99%

Problems on Graphs with Fixed Smallest Eigenvalue

Koolen

Yang

2020

Algebra Colloq.

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For strongly regular graphs with parameters (v, k, a, c) and fixed smallest eigenvalue −λ, Neumaier [5] gave two bounds of c by using algebraic property of strongly regular graphs. We define a new class of regular graphs called sesqui-regular graphs, which are amply regular graphs without parameter a. We prove that if a sesqui-regular graph * J.H. with parameters (v, k, c) has very large k and fixed smallest eigenvalue −λ, then c ≤ λ 2 (λ − 1) or v − k − 1 ≤ (λ−1) 2 4 + 1 holds.

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Graphs with least eigenvalue −2: Ten years on

Cited by 16 publications

References 126 publications

Non-backtracking Spectrum: Unitary Eigenvalues and Diagonalizability

Non-backtracking Spectrum: Unitary Eigenvalues and Diagonalizability

Eigenvalues of subgraphs of the cube

Problems on Graphs with Fixed Smallest Eigenvalue

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