Walks and the spectral radius of graphs

Nikiforov, Vladimir

doi:10.1016/j.laa.2006.02.003

Cited by 72 publications

(44 citation statements)

References 15 publications

(22 reference statements)

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“…In what follows, we use the following upper bound for λ k 1 (k being a positive integer) in terms of the clique number ω and the number of k-walks w k in G [10],…”

Section: Bounds For Spectral Gapmentioning

confidence: 99%

Trees with small spectral gap

2017

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Continuing the previous research, we consider trees with given number of vertices and minimal spectral gap. Using the computer search, we conjecture that this spectral invariant is minimized for double comet trees. The conjecture is confirmed for trees with at most 20 vertices; simultaneously no counterexamples are encountered. We provide theoretical results concerning double comets and putative trees that minimize the spectral gap. We also compare the spectral gap of regular graphs and paths. Finally, a sequence of inequalities that involve the same invariant is obtained.

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“…In what follows, we use the following upper bound for λ k 1 (k being a positive integer) in terms of the clique number ω and the number of k-walks w k in G [10],…”

Section: Bounds For Spectral Gapmentioning

confidence: 99%

Trees with small spectral gap

2017

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“…It seems difficult to give a constructive characterization of all matrices that force equality in (30), since the given condition is exact, but is too general for constructive characterization. Thus, we give just one construction, showing the great diversity of this class:…”

Section: Maximal Ky Fan Norms Of Matricesmentioning

confidence: 99%

Beyond graph energy: Norms of graphs and matrices

Nikiforov

2016

Linear Algebra and its Applications

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In 1978 Gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues, and ever since then graph energy has been intensively studied.Since graph energy is the trace norm of the adjacency matrix, matrix norms provide a natural background for its study. Thus, this paper surveys research on matrix norms that aims to expand and advance the study of graph energy.The focus is exclusively on the Ky Fan and the Schatten norms, both generalizing and enriching the trace norm. As it turns out, the study of extremal properties of these norms leads to numerous analytic problems with deep roots in combinatorics.The survey brings to the fore the exceptional role of Hadamard matrices, conference matrices, and conference graphs in matrix norms. In addition, a vast new matrix class is studied, a relaxation of symmetric Hadamard matrices.The survey presents solutions to just a fraction of a larger body of similar problems bonding analysis to combinatorics. Thus, open problems and questions are raised to outline topics for further investigation.

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“…It follows from the results in [6] that the right hand-side of the previous inequality is monotonically increasing with k. If, in addition, G is connected and not bipartite, it actually tends to 1 È n i=1 xi as k goes to infinity. We have provided upper and lower bounds for the maximum entry of the principal eigenvector of a connected graph that are sharp in some cases.…”

mentioning

confidence: 93%

Principal eigenvectors of irregular graphs

Cioabă¹,

Gregory²

2007

ELA

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Abstract. Let G be a connected graph. This paper studies the extreme entries of the principal eigenvector x of G, the unique positive unit eigenvector corresponding to the greatest eigenvalue λ 1 of the adjacency matrix of G. If G has maximum degree ∆, the greatest entry xmax of x is at most 1/ Õ 1 + λ 2 1 /∆. This improves a result of Papendieck and Recht. The least entry x min of x as well as the principal ratio xmax/x min are studied. It is conjectured that for connected graphs of order n ≥ 3, the principal ratio is always attained by one of the lollipop graphs obtained by attaching a path graph to a vertex of a complete graph.

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Walks and the spectral radius of graphs

Cited by 72 publications

References 15 publications

Trees with small spectral gap

Trees with small spectral gap

Beyond graph energy: Norms of graphs and matrices

Principal eigenvectors of irregular graphs

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