  Let $f(D(i, j), d_i, d_j)$ be a real function symmetric in $i$ and $j$ with
the property that $f(d, (1+o(1))np, (1+o(1))np)=(1+o(1))f(d, np, np)$ for
$d=1,2$. Let $G$ be a graph, $d_i$ denote the degree of a vertex $i$ of $G$ and
$D(i, j)$ denote the distance between vertices $i$ and $j$ in $G$. In this
paper, we define the $f$-weighted Laplacian matrix for random graphs in the
Erd$\ddot{o}$s-R$\acute{e}$nyi random graph model $\mathcal{G}_{n, p}$, where
$p\in (0, 1)$ is fixed. Four weighted Laplacian type energies: the weighted
Laplacian energy $\mathscr{LE}_f(G)$, weighted signless Laplacian energy
$\mathscr{LE}^{+}_f(G)$, weighted incidence energy $\mathscr{IE}_f(G)$ and the
weighted Laplacian-energy like invariant $\mathscr{LEL}_f(G)$ are introduced
and studied. We obtain the asymptotic values of $\mathscr{IE}_f(G)$ and
$\mathscr{LEL}_f(G)$, and the values of $\mathscr{LE}_f(G)$ and
$\mathscr{LE}_f^{+}(G)$ under the condition that $f(D(i, j), d_i, d_j)$ is a
function dependent only on $D(i, j)$. As a consequence, we get that for almost
all graphs $G_p\in \mathcal{G}_{n, p}$, the energy for the matrix with
degree-distance-based entries of $G_p$, $\mathscr{E}(W_f(G_p)) <
\mathscr{LE}_f(G_p),$ the Laplacian energy of the matrix, which is a
generalization of a conjecture by Gutman et al.
