Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On The Isoperimetric Spectrum of Graphs and Its Approximations   In this paper we consider higher isoperimetric numbers of a (finite directed)
graph. In this regard we focus on the $n$th mean isoperimetric constant of a
directed graph as the minimum of the mean outgoing normalized flows from a
given set of $n$ disjoint subsets of the vertex set of the graph. We show that
the second mean isoperimetric constant in this general setting, coincides with
(the mean version of) the classical Cheeger constant of the graph, while for
the rest of the spectrum we show that there is a fundamental difference between
the $n$th isoperimetric constant and the number obtained by taking the minimum
over all $n$-partitions. In this direction, we show that our definition is the
correct one in the sense that it satisfies a Federer-Fleming-type theorem, and
we also define and present examples for the concept of a supergeometric graph
as a graph whose mean isoperimetric constants are attained on partitions at all
levels. Moreover, considering the ${\bf NP}$-completeness of the isoperimetric
problem on graphs, we address ourselves to the approximation problem where we
prove general spectral inequalities that give rise to a general Cheeger-type
inequality as well. On the other hand, we also consider some algorithmic
aspects of the problem where we show connections to orthogonal representations
of graphs and following J.~Malik and J.~Shi ($2000$) we study the close
relationships to the well-known $k$-means algorithm and normalized cuts method.

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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: comp-gas
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The categories of this paper are: astro-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: adap-org
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The categories of this paper are: hep-lat
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The categories of this paper are: patt-sol
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The categories of this paper are: eess
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The categories of this paper are: q-bio
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The categories of this paper are: q-fin