Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Two problems on independent sets in graphs   Let $i_t(G)$ denote the number of independent sets of size $t$ in a graph
$G$. Levit and Mandrescu have conjectured that for all bipartite $G$ the
sequence $(i_t(G))_{t \geq 0}$ (the {\em independent set sequence} of $G$) is
unimodal. We provide evidence for this conjecture by showing that is true for
almost all equibipartite graphs. Specifically, we consider the random
equibipartite graph $G(n,n,p)$, and show that for any fixed $p\in(0,1]$ its
independent set sequence is almost surely unimodal, and moreover almost surely
log-concave except perhaps for a vanishingly small initial segment of the
sequence. We obtain similar results for $p=\tilde{\Omega}(n^{-1/2})$.
  We also consider the problem of estimating $i(G)=\sum_{t \geq 0} i_t(G)$ for
$G$ in various families. We give a sharp upper bound on the number of
independent sets in an $n$-vertex graph with minimum degree $\delta$, for all
fixed $\delta$ and sufficiently large $n$. Specifically, we show that the
maximum is achieved uniquely by $K_{\delta, n-\delta}$, the complete bipartite
graph with $\delta$ vertices in one partition class and $n-\delta$ in the
other.
  We also present a weighted generalization: for all fixed $x>0$ and $\delta
>0$, as long as $n=n(x,\delta)$ is large enough, if $G$ is a graph on $n$
vertices with minimum degree $\delta$ then $\sum_{t \geq 0} i_t(G)x^t \leq
\sum_{t \geq 0} i_t(K_{\delta, n-\delta})x^t$ with equality if and only if
$G=K_{\delta, n-\delta}$.

8
The categories of this paper are: hep-ex
28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
0
The categories of this paper are: hep-ph
5
The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
19
The categories of this paper are: eess
17
The categories of this paper are: q-fin