10 research outputs found
Faber-Krahn Type Inequalities for Trees
The Faber-Krahn theorem states that among all bounded domains with the same
volume in (with the standard Euclidean metric), a ball that has
lowest first Dirichlet eigenvalue. Recently it has been shown that a similar
result holds for (semi-)regular trees. In this article we show that such a
theorem also hold for other classes of (not necessarily non-regular) trees.
However, for these new results no couterparts in the world of the
Laplace-Beltrami-operator on manifolds are known.Comment: 19 pages, 5 figure
Largest Laplacian Eigenvalue and Degree Sequences of Trees
We investigate the structure of trees that have greatest maximum eigenvalue
among all trees with a given degree sequence. We show that in such an extremal
tree the degree sequence is non-increasing with respect to an ordering of the
vertices that is obtained by breadth-first search. This structure is uniquely
determined up to isomorphism. We also show that the maximum eigenvalue in such
classes of trees is strictly monotone with respect to majorization.Comment: 9 pages, 2 figure
Graphs with Given Degree Sequence and Maximal Spectral Radius
We describe the structure of those graphs that have largest spectral radius
in the class of all connected graphs with a given degree sequence. We show that
in such a graph the degree sequence is non-increasing with respect to an
ordering of the vertices induced by breadth-first search. For trees the
resulting structure is uniquely determined up to isomorphism. We also show that
the largest spectral radius in such classes of trees is strictly monotone with
respect to majorization.Comment: 12 pages, 4 figures; revised version. Important change: Theorem 3
(formely Theorem 7) now states (and correctly proofs) the majorization result
only for "degree sequences of trees" (instead for general connected graphs).
Bo Zhou from the South China Normal University in Guangzhou, P.R. China, has
found a counter-example to the stronger resul
The first Dirichlet eigenvalue of bicyclic graphs
summary:In this paper, we have investigated some properties of the first Dirichlet eigenvalue of a bicyclic graph with boundary condition. These results can be used to characterize the extremal bicyclic graphs with the smallest first Dirichlet eigenvalue among all the bicyclic graphs with a given graphic bicyclic degree sequence with minor conditions. Moreover, the extremal bicyclic graphs with the smallest first Dirichlet eigenvalue among all the bicycle graphs with fixed interior vertices of degree at least 3 are obtained
Eigenvectors of the discrete Laplacian on regular graphs - a statistical approach
In an attempt to characterize the structure of eigenvectors of random regular
graphs, we investigate the correlations between the components of the
eigenvectors associated to different vertices. In addition, we provide
numerical observations, suggesting that the eigenvectors follow a Gaussian
distribution. Following this assumption, we reconstruct some properties of the
nodal structure which were observed in numerical simulations, but were not
explained so far. We also show that some statistical properties of the nodal
pattern cannot be described in terms of a percolation model, as opposed to the
suggested correspondence for eigenvectors of 2 dimensional manifolds.Comment: 28 pages, 11 figure
Geometric representations and symmetries of graphs, maps and other discrete structures and applications in science
Bu proje çalışmasının birinci amacı çizgelerin özdeğer ve özvektör yapılarını çizge özellik ve sabitleri ile ilişkilendirmektir. İkinci amacı biyoloji, bioinformatik, dinamik sistemler, haberleşme, kriptoloji ve sosyal ağlar gibi bir birinden çok farklı alanlardan gelen birbirinden tamamen bağımsız olan temel problemler için çizge kuramı ile özgün modellenmesi ve ortaya çıkan çizge problemlerin çözülmesidir. Üçüncü amacı çizgelerin Castelnuovo-Mumford regülaritesine indirgenmiş eşleşme sayısı ile üstten etkin sınırlar getirmektir.TÜBİTAKPublisher's Versio