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Let $f(D(i, j), d_i, d_j)$ be a real function symmetric in $i$ and $j$ withthe 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 thispaper, we define the $f$-weighted Laplacian matrix for random graphs in theErd$\ddot{o}$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 weightedLaplacian energy $\mathscr{LE}_f(G)$, weighted signless Laplacian energy$\mathscr{LE}^{+}_f(G)$, weighted incidence energy $\mathscr{IE}_f(G)$ and theweighted Laplacian-energy like invariant $\mathscr{LEL}_f(G)$ are introducedand 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 afunction dependent only on $D(i, j)$. As a consequence, we get that for almostall graphs $G_p\in \mathcal{G}_{n, p}$, the energy for the matrix withdegree-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 ageneralization of a conjecture by Gutman et al.

Asymptotic values of four Laplacian-type energies for matrices with degree-distance-based entries of random graphs