  We determine the size of $k$-core in a large class of dense graph sequences.
Let $G_n$ be a sequence of undirected, $n$-vertex graphs with edge weights
$\{a^n_{i,j}\}_{i,j \in [n]}$ that converges to a kernel $W:[0,1]^2\to
[0,+\infty)$ in the cut metric. Keeping an edge $(i,j)$ of $G_n$ with
probability $\min \{ {a^n_{i,j}}/{n},1 \}$ independently, we obtain a sequence
of random graphs $G_n(\frac{1}{n})$. Denote by $C_k(G)$ the size of $k$-core in
graph $G$, by $X^W$ the branching process associated with the kernel $W$, by
$\mathcal{A}$ the property of a branching process that the initial particle has
at least $k$ children, each of which has at least $k-1$ children, each of which
has at least $k-1$ children, and so on. Using branching process and theory of
dense graph limits, under mild assumptions we obtain the size of $k$-core of
random graphs $G_n(\frac{1}{n})$, \begin{align*}
C_k\left(G_n\left(\frac{1}{n}\right)\right) =n
\mathbb{P}_{X^W}\left(\mathcal{A}\right) +o_p(n). \end{align*} Our result can
also be used to obtain the threshold of appearance of a $k$-core of order $n$.
In addition, we obtain a probabilistic result concerning cut-norm and branching
process which might be of independent interest.
