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

Abstract

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 ${\mathcal{LE}}_f(G)$, weighted signless Laplacian energy ${\mathcal{LE}}_f^{+}(G)$, weighted incidence energy ${\mathcal{IE}}_f(G)$ and the weighted Laplacian-energy like invariant ${\mathcal{LEL}}_f(G)$ are introduced and studied. We obtain the asymptotic values of ${\mathcal{IE}}_f(G)$ and ${\mathcal{LEL}}_f(G)$, and the values of ${\mathcal{LE}}_f(G)$ and ${\mathcal{LE}}_f^{+}(G)$ 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 $G_p$, $\mathcal{E}(W_f(G_p))<{\mathcal{LE}}_f(G_p),$ the Laplacian energy of the matrix, which is a generalization of a conjecture by Gutman et al.

18 pages