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,…,vn}, without isolated vertices and sequence of vertex degrees Δ=d1≥d2≥⋯≥dn=δ>0, di=dG(vi). If the vertices vi and vj are adjacent, we denote it as vivj∈E(G) or i∼j. With TI we denote a topological index that can be represented as TI=TI(G)=∑i∼jF(di,dj), where F is an appropriately chosen function with the property F(x,y)=F(y,x). A general extended adjacency matrix A=(aij) of G is defined as aij=F(di,dj) if the vertices vi and vj are adjacent, and aij=0 otherwise. Denote by fi, i=1,2,…,n the eigenvalues of A. The “energy” of the general extended adjacency matrix is defined as ETI=ETI(G)=∑i=1n|fi|. Lower and upper bounds on ETI are obtained. By means of the present approach a plethora of earlier established results can be obtained as special cases.