Suppose that the vertex set of a connected graph G is $$V(G)=\{v_1,\ldots ,v_n\}$$ . Then we denote by $$Tr_{G}(v_i)$$ sum distances between $$v_i$$ and all other vertices G. Let Tr(G) be $$n\times n$$ diagonal matrix with its (i, i)-entry equal to $$Tr_{G}(v_{i})$$ D(G) distance $$Q_{D}(G)=Tr(G)+D(G)$$ signless Laplacian The largest eigenvalues $$Q_D(G)$$ called spectral radius In this ...

Let f(D(i,j),di,dj) 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. G graph, di denote degree of vertex D(i,j) distance between vertices G. In this paper, we define f-weighted Laplacian matrix random graphs Erdös-Rényi graph model Gn,p, where p?(0,1) is fixed. Four weighted type energies: energy LEf(G), signless LEf+(G), in...