Degree-based energies of graphs

Let G=(V,E) be a simple graph of order n and size m, with vertex set V(G)={v₁,v₂,…,v_n}, without isolated vertices and sequence of vertex degrees Δ=d₁≥d₂≥⋯≥d_n=δ>0, d_i=d_G(v_i). If the vertices v_i and v_j are adjacent, we denote it as v_iv_j∈E(G) or i∼j. With TI we denote a topological index that can be represented as TI=TI(G)=∑_i∼jF(d_i,d_j), where F is an appropriately chosen function with the property F(x,y)=F(y,x). A general extended adjacency matrix A=(a_ij) of G is defined as a_ij=F(d_i,d_j) if the vertices v_i and v_j are adjacent, and a_ij=0 otherwise. Denote by f_i, i=1,2,…,n the eigenvalues of A. The “energy” of the general extended adjacency matrix is defined as E_TI=E_TI(G)=∑_i=1 ⁿ|f_i|. Lower and upper bounds on E_TI are obtained. By means of the present approach a plethora of earlier established results can be obtained as special cases.