5 research outputs found
Algebraic Connectivity and Degree Sequences of Trees
We investigate the structure of trees that have minimal algebraic
connectivity among all trees with a given degree sequence. We show that such
trees are caterpillars and that the vertex degrees are non-decreasing on every
path on non-pendant vertices starting at the characteristic set of the Fiedler
vector.Comment: 8 page
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
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
Synchronization of networks with prescribed degree distributions
We show that the degree distributions of graphs do not suffice to
characterize the synchronization of systems evolving on them. We prove that,
for any given degree sequence satisfying certain conditions, there exists a
connected graph having that degree sequence for which the first nontrivial
eigenvalue of the graph Laplacian is arbitrarily close to zero. Consequently,
complex dynamical systems defined on such graphs have poor synchronization
properties. The result holds under quite mild assumptions, and shows that there
exists classes of random, scale-free, regular, small-world, and other common
network architectures which impede synchronization. The proof is based on a
construction that also serves as an algorithm for building non-synchronizing
networks having a prescribed degree distribution.Comment: v2: A new theorem and a numerical example added. To appear in IEEE
Trans. Circuits and Systems I: Fundamental Theory and Application
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