General Randić matrix and general Randić incidence matrix

Let GG be a connected graph with vertex set V(G)={v1,…,vn}V(G)={v1,…,vn} and edge set E(G)={e1,…,em}E(G)={e1,…,em}. Let didi be the degree of the vertex vivi. The general Randić matrix Rα=((Rα)ij)n×n of GG is defined by (Rα)ij=(didj)α if vertices vivi and vjvj are adjacent in GG and 0 otherwise. The Randić signless Laplacian matrix Qα=D2α+1+Rα, where αα is a nonzero real number and DD is the degree diagonal matrix of GG. The general Randić energy REαREα is the sum of absolute values of the eigenvalues of Rα. The general Randić incidence matrix BRα=((BRα)ij)n×mBRα=((BRα)ij)n×m of a graph GG is defined by (BRα)ij=diα if vivi is incident to ejej and 0 otherwise. Naturally, the general Randić incidence energy BEαBEα is the sum of the singular values of BRαBRα. In this paper, we investigate the connected graphs with ss distinct Rα-eigenvalues, where 2≤s≤n2≤s≤n. Moreover, we establish the relation between the Randić signless Laplacian eigenvalues of GG and the general Randić energy of its subdivided graph S(G)S(G). Also we give lower and upper bounds on the general Randić incidence energy. Finally, the general Randić incidence energy of a graph and that of its subgraphs are compared.