The asymptotic value of energy for matrices with degree-distance-based entries of random graphs

Abstract

For a graph $G=(V,E)$ and $i,j\in V$, denote the distance between $i$ and $j$ in $G$ by $D(i,j)$ and the degrees of $i$, $j$ by $d_i$, $d_j$, respectively. Let $f(D(i,j),d_i,d_j)$ be a function symmetric in $i$ and $j$. Define a matrix $W_f(G)$, called the weighted distance matrix, of $G$, with the $ij$-entry $W_f(G)(i,j)=f(D(i,j),d_i,d_j)$ if $i\ne j$ and $W_f(G)(i,j)=0$ if $i=j$. In this paper, we prove that if the symmetric function $f$ satisfies that $f(D(i,j),(1+o(1))np,(1+o(1))np)=(1+o(1))f(D(i,j),np,np)$, then for almost all graphs $G_p$ in the $Erd\ddot{o}s$-$R\acute{e}nyi$ random graph model ${\mathcal{G}}_{n,p}$, the energy of $W_f(G_p)$ is $\{(\frac{8}{3\pi }\sqrt{p(1-p)}+o(1))\cdot |f(1,np,np)-f(2,np,np)|+o(|f(2,np,np)|)\}\cdot n^{3/2}$. As a consequence, we give the asymptotic values of energies of a variety of weighted distance matrices with function $f$ from distance-based only and mixed with degree-distance-based topological indices of chemical use. This generalizes our former result with only degree-based weights.

15 pages