1,232 research outputs found
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
Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends
We study the following classes of beyond-planar graphs: 1-planar, IC-planar,
and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar,
and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every
edge is crossed at most once. A 1-planar drawing is IC-planar if no two pairs
of crossing edges share a vertex. A 1-planar drawing is NIC-planar if no two
pairs of crossing edges share two vertices. We study the relations of these
beyond-planar graph classes (beyond-planar graphs is a collective term for the
primary attempts to generalize the planar graphs) to right-angle crossing (RAC)
graphs that admit compact drawings on the grid with few bends. We present four
drawing algorithms that preserve the given embeddings. First, we show that
every -vertex NIC-planar graph admits a NIC-planar RAC drawing with at most
one bend per edge on a grid of size . Then, we show that
every -vertex 1-planar graph admits a 1-planar RAC drawing with at most two
bends per edge on a grid of size . Finally, we make two
known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at
most one bend per edge and for drawing IC-planar RAC graphs straight-line
Dimension of posets with planar cover graphs excluding two long incomparable chains
It has been known for more than 40 years that there are posets with planar
cover graphs and arbitrarily large dimension. Recently, Streib and Trotter
proved that such posets must have large height. In fact, all known
constructions of such posets have two large disjoint chains with all points in
one chain incomparable with all points in the other. Gutowski and Krawczyk
conjectured that this feature is necessary. More formally, they conjectured
that for every , there is a constant such that if is a poset
with a planar cover graph and excludes , then
. We settle their conjecture in the affirmative. We also discuss
possibilities of generalizing the result by relaxing the condition that the
cover graph is planar.Comment: New section on connections with graph minors, small correction
Join-Reachability Problems in Directed Graphs
For a given collection G of directed graphs we define the join-reachability
graph of G, denoted by J(G), as the directed graph that, for any pair of
vertices a and b, contains a path from a to b if and only if such a path exists
in all graphs of G. Our goal is to compute an efficient representation of J(G).
In particular, we consider two versions of this problem. In the explicit
version we wish to construct the smallest join-reachability graph for G. In the
implicit version we wish to build an efficient data structure (in terms of
space and query time) such that we can report fast the set of vertices that
reach a query vertex in all graphs of G. This problem is related to the
well-studied reachability problem and is motivated by emerging applications of
graph-structured databases and graph algorithms. We consider the construction
of join-reachability structures for two graphs and develop techniques that can
be applied to both the explicit and the implicit problem. First we present
optimal and near-optimal structures for paths and trees. Then, based on these
results, we provide efficient structures for planar graphs and general directed
graphs
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
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