Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The independent set sequence of regular bipartite graphs   Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$.
Alavi, Erd\H{o}s, Malde and Schwenk made the conjecture that if $G$ is a tree
then the independent set sequence $\{i_t(G)\}_{t\geq 0}$ of $G$ is unimodal;
Levit and Mandrescu further conjectured that this should hold for all bipartite
$G$.
  We consider the independent set sequence of finite regular bipartite graphs,
and graphs obtained from these by percolation (independent deletion of edges).
Using bounds on the independent set polynomial $P(G,\lambda):=\sum_{t \geq 0}
i_t(G)\lambda^t$ for these graphs, we obtain partial unimodality results in
these cases.
  We then focus on the discrete hypercube $Q_d$, the graph on vertex set
$\{0,1\}^d$ with two strings adjacent if they differ on exactly one coordinate.
We obtain asymptotically tight estimates for $i_{t(d)}(Q_d)$ in the range
$t(d)/2^{d-1} > 1-1/\sqrt{2}$, and nearly matching upper and lower bounds
otherwise. We use these estimates to obtain a stronger partial unimodality
result for the independent set sequence of $Q_d$.

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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: nlin
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The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: q-bio
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The categories of this paper are: physics
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The categories of this paper are: stat
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The categories of this paper are: funct-an
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The categories of this paper are: atom-ph