Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Minimizing the number of 5-cycles in graphs with given edge-density   Motivated by the work of Razborov about the minimal density of triangles in
graphs we study the minimal density of the 5-cycle $C_5$. We show that every
graph of order $n$ and size $\left( 1-\frac{1}{k}\right)\binom{n}{2}$, where
$k\ge 3$ is an integer, contains at least \[ \left( \frac{1}{10} -\frac{1}{2k}
+ \frac{1}{k^2} - \frac{1}{k^3} + \frac{2}{5 k^4} \right)n^5 +o(n^5) \] copies
of $C_5$. This bound is optimal, since a matching upper bound is given by the
balanced complete $k$-partite graph. The proof is based on the flag algebras
framework. We also provide a stability result. An SDP solver is not necessary
to verify our proofs.

18
The categories of this paper are: econ
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
24
The categories of this paper are: dg-ga
8
The categories of this paper are: hep-ex
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
17
The categories of this paper are: q-fin
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys