13,370 research outputs found
Bounds on the eigenvalues of graphs with cut vertices or edges
AbstractIn this paper, we characterize the extremal graph having the maximal Laplacian spectral radius among the connected bipartite graphs with n vertices and k cut vertices, and describe the extremal graph having the minimal least eigenvalue of the adjacency matrices of all the connected graphs with n vertices and k cut edges. We also present lower bounds on the least eigenvalue in terms of the number of cut vertices or cut edges and upper bounds on the Laplacian spectral radius in terms of the number of cut vertices
A theory of spectral partitions of metric graphs
We introduce an abstract framework for the study of clustering in metric
graphs: after suitably metrising the space of graph partitions, we restrict
Laplacians to the clusters thus arising and use their spectral gaps to define
several notions of partition energies; this is the graph counterpart of the
well-known theory of spectral minimal partitions on planar domains and includes
the setting in [Band \textit{et al}, Comm.\ Math.\ Phys.\ \textbf{311} (2012),
815--838] as a special case. We focus on the existence of optimisers for a
large class of functionals defined on such partitions, but also study their
qualitative properties, including stability, regularity, and parameter
dependence. We also discuss in detail their interplay with the theory of nodal
partitions. Unlike in the case of domains, the one-dimensional setting of
metric graphs allows for explicit computation and analytic -- rather than
numerical -- results. Not only do we recover the main assertions in the theory
of spectral minimal partitions on domains, as studied in [Conti \textit{et al},
Calc.\ Var.\ \textbf{22} (2005), 45--72; Helffer \textit{et al}, Ann.\ Inst.\
Henri Poincar\'e Anal.\ Non Lin\'eaire \textbf{26} (2009), 101--138], but we
can also generalise some of them and answer (the graph counterparts of) a few
open questions
Approximating the Spectrum of a Graph
The spectrum of a network or graph with adjacency matrix ,
consists of the eigenvalues of the normalized Laplacian . This set of eigenvalues encapsulates many aspects of the structure
of the graph, including the extent to which the graph posses community
structures at multiple scales. We study the problem of approximating the
spectrum , of in the regime where the graph is too
large to explicitly calculate the spectrum. We present a sublinear time
algorithm that, given the ability to query a random node in the graph and
select a random neighbor of a given node, computes a succinct representation of
an approximation , such that . Our algorithm has query complexity and running time ,
independent of the size of the graph, . We demonstrate the practical
viability of our algorithm on 15 different real-world graphs from the Stanford
Large Network Dataset Collection, including social networks, academic
collaboration graphs, and road networks. For the smallest of these graphs, we
are able to validate the accuracy of our algorithm by explicitly calculating
the true spectrum; for the larger graphs, such a calculation is computationally
prohibitive.
In addition we study the implications of our algorithm to property testing in
the bounded degree graph model
Spectral Bounds for the Connectivity of Regular Graphs with Given Order
The second-largest eigenvalue and second-smallest Laplacian eigenvalue of a
graph are measures of its connectivity. These eigenvalues can be used to
analyze the robustness, resilience, and synchronizability of networks, and are
related to connectivity attributes such as the vertex- and edge-connectivity,
isoperimetric number, and characteristic path length. In this paper, we present
two upper bounds for the second-largest eigenvalues of regular graphs and
multigraphs of a given order which guarantee a desired vertex- or
edge-connectivity. The given bounds are in terms of the order and degree of the
graphs, and hold with equality for infinite families of graphs. These results
answer a question of Mohar.Comment: 24 page
Towards a better approximation for sparsest cut?
We give a new -approximation for sparsest cut problem on graphs
where small sets expand significantly more than the sparsest cut (sets of size
expand by a factor bigger, for some small ; this
condition holds for many natural graph families). We give two different
algorithms. One involves Guruswami-Sinop rounding on the level- Lasserre
relaxation. The other is combinatorial and involves a new notion called {\em
Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which
we show exists in the input graph. Both algorithms run in time . We also show similar approximation algorithms in graphs with
genus with an analogous local expansion condition. This is the first
algorithm we know of that achieves -approximation on such general
family of graphs
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