Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
$D$ cannot exceed $2\sqrt{D-1}$. 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 $\rho(G)$ of a planar graph $G$ of maximum vertex degree
$D\ge 4$ satisfies $\sqrt{D}\le \rho(G)\le \sqrt{8D-16}+7.75$. This result is
best possible up to the additive constant--we construct an (infinite) planar
graph of maximum degree $D$, whose spectral radius is $\sqrt{8D-16}$. 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 $k$, these bounds can be improved by excluding $K_{2,k}$ 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 $\{p,q\}$ 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.

29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
9
The categories of this paper are: nlin
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas