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Comparison of Metric Spectral Gaps

Assaf Naor (2014)

Analysis and Geometry in Metric Spaces

Let A = (aij) ∊ Mn(ℝ) be an n by n symmetric stochastic matrix. For p ∊ [1, ∞) and a metric space (X, dX), let γ(A, dpx) be the infimum over those γ ∊ (0,∞] for which every x1, . . . , xn ∊ X satisfy [...] Thus γ (A, dpx) measures the magnitude of the nonlinear spectral gap of the matrix A with respect to the kernel dpX : X × X →[0,∞). We study pairs of metric spaces (X, dX) and (Y, dY ) for which there exists Ψ: (0,∞)→(0,∞) such that γ (A, dpX) ≤Ψ (A, dpY ) for every symmetric stochastic A ∊ Mn(ℝ)...

Connectivity, toughness, spanning trees of bounded degree, and the spectrum of regular graphs

Sebastian M. Cioabă, Xiaofeng Gu (2016)

Czechoslovak Mathematical Journal

The eigenvalues of graphs are related to many of its combinatorial properties. In his fundamental work, Fiedler showed the close connections between the Laplacian eigenvalues and eigenvectors of a graph and its vertex-connectivity and edge-connectivity. We present some new results describing the connections between the spectrum of a regular graph and other combinatorial parameters such as its generalized connectivity, toughness, and the existence of spanning trees with bounded degree.

Constructions for type I trees with nonisomorphic Perron branches

Stephen J. Kirkland (1999)

Czechoslovak Mathematical Journal

A tree is classified as being type I provided that there are two or more Perron branches at its characteristic vertex. The question arises as to how one might construct such a tree in which the Perron branches at the characteristic vertex are not isomorphic. Motivated by an example of Grone and Merris, we produce a large class of such trees, and show how to construct others from them. We also investigate some of the properties of a subclass of these trees. Throughout, we exploit connections between...

Controllable graphs

D. Cvetković, P. Rowlinson, Z. Stanić, M. G. Yoon (2011)

Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques

Cover matrices of posets and their spectra

Milica Anđelić, C. M. da Fonseca (2009)

Czechoslovak Mathematical Journal

We analyze the spectra of the cover matrix of a given poset. Some consequences on the multiplicities are provided.

Covering energy of posets and its bounds

Vandana P. Bhamre, Madhukar M. Pawar (2023)

Mathematica Bohemica

The concept of covering energy of a poset is known and its McClelland type bounds are available in the literature. In this paper, we establish formulas for the covering energy of a crown with 2 n elements and a fence with n elements. A lower bound for the largest eigenvalue of a poset is established. Using this lower bound, we improve the McClelland type bounds for the covering energy for some special classes of posets.

Cut-off for large sums of graphs

Bernard Ycart (2007)

Annales de l’institut Fourier

If L is the combinatorial Laplacian of a graph, exp ( - L t ) converges to a matrix with identical coefficients. The speed of convergence is measured by the maximal entropy distance. When the graph is the sum of a large number of components, a cut-off phenomenon may occur: before some instant the distance to equilibrium tends to infinity; after that instant it tends to 0 . A sufficient condition for cut-off is given, and the cut-off instant is expressed as a function of the gap and eigenvectors of components....

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