199,486 research outputs found
Bounds on the radius and status of graphs
Two classical concepts of centrality in a graph are the median and the
center. The connected notions of the status and the radius of a graph seem to
be in no relation. In this paper, however, we show a clear connection of both
concepts, as they obtain their minimum and maximum values at the same type of
tree graphs. Trees with fixed maximum degree and extremum radius and status,
resp., are characterized. The bounds on radius and status can be transferred to
general connected graphs via spanning trees.
A new method of proof allows not only to regain results of Lin et al. on
graphs with extremum status, but it allows also to prove analogous results on
graphs with extremum radius
Sharp Bounds for the Signless Laplacian Spectral Radius in Terms of Clique Number
In this paper, we present a sharp upper and lower bounds for the signless
Laplacian spectral radius of graphs in terms of clique number. Moreover, the
extremal graphs which attain the upper and lower bounds are characterized. In
addition, these results disprove the two conjectures on the signless Laplacian
spectral radius in [P. Hansen and C. Lucas, Bounds and conjectures for the
signless Laplacian index of graphs, Linear Algebra Appl., 432(2010) 3319-3336].Comment: 15 pages 1 figure; linear algebra and its applications 201
Spectral radius of finite and infinite planar graphs and of graphs of bounded genus
It is well known that the spectral radius of a tree whose maximum degree is
cannot exceed . In this paper we derive similar bounds for
arbitrary planar graphs and for graphs of bounded genus. It is proved that a
the spectral radius of a planar graph of maximum vertex degree
satisfies . This result is
best possible up to the additive constant--we construct an (infinite) planar
graph of maximum degree , whose spectral radius is . This
generalizes and improves several previous results and solves an open problem
proposed by Tom Hayes. Similar bounds are derived for graphs of bounded genus.
For every , these bounds can be improved by excluding as a
subgraph. In particular, the upper bound is strengthened for 5-connected
graphs. All our results hold for finite as well as for infinite graphs.
At the end we enhance the graph decomposition method introduced in the first
part of the paper and apply it to tessellations of the hyperbolic plane. We
derive bounds on the spectral radius that are close to the true value, and even
in the simplest case of regular tessellations of type we derive an
essential improvement over known results, obtaining exact estimates in the
first order term and non-trivial estimates for the second order asymptotics
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