6,519 research outputs found
Bounds on the maximum multiplicity of some common geometric graphs
We obtain new lower and upper bounds for the maximum multiplicity of some
weighted and, respectively, non-weighted common geometric graphs drawn on n
points in the plane in general position (with no three points collinear):
perfect matchings, spanning trees, spanning cycles (tours), and triangulations.
(i) We present a new lower bound construction for the maximum number of
triangulations a set of n points in general position can have. In particular,
we show that a generalized double chain formed by two almost convex chains
admits {\Omega}(8.65^n) different triangulations. This improves the bound
{\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by
Aichholzer et al.
(ii) We present a new lower bound of {\Omega}(12.00^n) for the number of
non-crossing spanning trees of the double chain composed of two convex chains.
The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years.
(iii) Using a recent upper bound of 30^n for the number of triangulations,
due to Sharir and Sheffer, we show that n points in the plane in general
position admit at most O(68.62^n) non-crossing spanning cycles.
(iv) We derive lower bounds for the number of maximum and minimum weighted
geometric graphs (matchings, spanning trees, and tours). We show that the
number of shortest non-crossing tours can be exponential in n. Likewise, we
show that both the number of longest non-crossing tours and the number of
longest non-crossing perfect matchings can be exponential in n. Moreover, we
show that there are sets of n points in convex position with an exponential
number of longest non-crossing spanning trees. For points in convex position we
obtain tight bounds for the number of longest and shortest tours. We give a
combinatorial characterization of the longest tours, which leads to an O(nlog
n) time algorithm for computing them
Counting Carambolas
We give upper and lower bounds on the maximum and minimum number of geometric
configurations of various kinds present (as subgraphs) in a triangulation of
points in the plane. Configurations of interest include \emph{convex
polygons}, \emph{star-shaped polygons} and \emph{monotone paths}. We also
consider related problems for \emph{directed} planar straight-line graphs.Comment: update reflects journal version, to appear in Graphs and
Combinatorics; 18 pages, 13 figure
Exact controllability of multiplex networks
Date of Acceptance: 11/09/2014Peer reviewedPublisher PD
Convex Graph Invariant Relaxations For Graph Edit Distance
The edit distance between two graphs is a widely used measure of similarity
that evaluates the smallest number of vertex and edge deletions/insertions
required to transform one graph to another. It is NP-hard to compute in
general, and a large number of heuristics have been proposed for approximating
this quantity. With few exceptions, these methods generally provide upper
bounds on the edit distance between two graphs. In this paper, we propose a new
family of computationally tractable convex relaxations for obtaining lower
bounds on graph edit distance. These relaxations can be tailored to the
structural properties of the particular graphs via convex graph invariants.
Specific examples that we highlight in this paper include constraints on the
graph spectrum as well as (tractable approximations of) the stability number
and the maximum-cut values of graphs. We prove under suitable conditions that
our relaxations are tight (i.e., exactly compute the graph edit distance) when
one of the graphs consists of few eigenvalues. We also validate the utility of
our framework on synthetic problems as well as real applications involving
molecular structure comparison problems in chemistry.Comment: 27 pages, 7 figure
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Geometric aspects of 2-walk-regular graphs
A -walk-regular graph is a graph for which the number of walks of given
length between two vertices depends only on the distance between these two
vertices, as long as this distance is at most . Such graphs generalize
distance-regular graphs and -arc-transitive graphs. In this paper, we will
focus on 1- and in particular 2-walk-regular graphs, and study analogues of
certain results that are important for distance regular graphs. We will
generalize Delsarte's clique bound to 1-walk-regular graphs, Godsil's
multiplicity bound and Terwilliger's analysis of the local structure to
2-walk-regular graphs. We will show that 2-walk-regular graphs have a much
richer combinatorial structure than 1-walk-regular graphs, for example by
proving that there are finitely many non-geometric 2-walk-regular graphs with
given smallest eigenvalue and given diameter (a geometric graph is the point
graph of a special partial linear space); a result that is analogous to a
result on distance-regular graphs. Such a result does not hold for
1-walk-regular graphs, as our construction methods will show
Non-Weyl Resonance Asymptotics for Quantum Graphs
We consider the resonances of a quantum graph that consists of a
compact part with one or more infinite leads attached to it. We discuss the
leading term of the asymptotics of the number of resonances of in
a disc of a large radius. We call a \emph{Weyl graph} if the
coefficient in front of this leading term coincides with the volume of the
compact part of . We give an explicit topological criterion for a
graph to be Weyl. In the final section we analyze a particular example in some
detail to explain how the transition from the Weyl to the non-Weyl case occurs.Comment: 29 pages, 2 figure
On the Fiedler value of large planar graphs
The Fiedler value , also known as algebraic connectivity, is the
second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler
value among all planar graphs with vertices, denoted by
, and we show the bounds . We also provide bounds on the maximum
Fiedler value for the following classes of planar graphs: Bipartite planar
graphs, bipartite planar graphs with minimum vertex degree~3, and outerplanar
graphs. Furthermore, we derive almost tight bounds on for two
more classes of graphs, those of bounded genus and -minor-free graphs.Comment: 21 pages, 4 figures, 1 table. Version accepted in Linear Algebra and
Its Application
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