Asymptotic values of four Laplacian-type energies for matrices with degree-distance-based entries of random graphs (Q2228530)

This paper studies asymptotic values of Laplacian-type energies for matrices with degree-distance-based entries of the Erdös-Rényi random graph \(G(n,p)\). Let a real symmetric function \(f\) be given over \(G\), let \(\operatorname{LEL}_f(G)\) be the associated weighted Laplacian-energy like invariant and \(\operatorname{IE}_f(G)\) be the weighted incidence energy. Let \(f_1(d_i,d_j)\) and \(f_2(d_i,d_j)\) be two symmetric functions satisfying \(f_1((1+o(1))np,(1+o(1))np)=(1+o(1))f_1(np,np)\) and \(f_s((1+o(1))np,(1+o(1))np)=(1+o(1))f_2(np,np)\), then it is shown that almost surely \begin{itemize} \item[(i)] if \(f_1(np,np)/f_2(np,np)\rightarrow\infty\) or \(f_1(np,np)/f_2(np,np)\rightarrow-\infty\), then \(\operatorname{LEL}_f(G(n,p))=\sqrt{|f_1(np,np)|}(\sqrt{p}+o(1))n^{3/2}\) and \(\operatorname{IE}_f(G(n,p))=\sqrt{|f_1(np,np)|}(\sqrt{p}+o(1))n^{3/2}\); \item[(ii)] if \(f_1(np,np)/f_2(np,np)\rightarrow C\) for some constant \(C\), then \(\operatorname{LEL}_f(G(n,p))=\sqrt{|f_2(np,np)|}\) \((\sqrt{1+(C-1)p}+o(1))\) \(n^{3/2}\) and \(\operatorname{LEL}_f(G(n,p))=\sqrt{|f_2(np,np)|}\) \((\sqrt{1+(C-1)p}+o(1))\) \(n^{3/2}\). \end{itemize}