  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 $\mathcal{A}$ 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 the 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*} \text{size of $k$-core of }
G_n\left(\frac{1}{n}\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 $n$.
