1,205 research outputs found
Spectrally degenerate graphs: Hereditary case
It is well known that the spectral radius of a tree whose maximum degree is D
cannot exceed 2sqrt{D-1}. Similar upper bound holds for arbitrary planar
graphs, whose spectral radius cannot exceed sqrt{8D}+10, and more generally,
for all d-degenerate graphs, where the corresponding upper bound is sqrt{4dD}.
Following this, we say that a graph G is spectrally d-degenerate if every
subgraph H of G has spectral radius at most sqrt{d.Delta(H)}. In this paper we
derive a rough converse of the above-mentioned results by proving that each
spectrally d-degenerate graph G contains a vertex whose degree is at most
4dlog_2(D/d) (if D>=2d). It is shown that the dependence on D in this upper
bound cannot be eliminated, as long as the dependence on d is subexponential.
It is also proved that the problem of deciding if a graph is spectrally
d-degenerate is co-NP-complete.Comment: Updated after reviewer comments. 14 pages, no figure
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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